1. Introduction
In present-day astrophysics, modified gravitational frameworks (MGFs) have emerged as one of the most fascinating and viable candidates of dark energy (DE)—an unusual form of energy which causes our cosmos to expand [1, 2]. It is argued that MGFs can explain various cosmic issues successfully which the general relativity theory (GR) has failed to answer. In the literature [3], several noteworthy proposals for modifying the curvature sector of the Lagrangian of the GR framework have been put forward by numerous researchers. Some examples of paramount importance include the f(R) theory [4]; the scalar-tensor theories, in particular, Brans-Dicke theory [5, 6] and Saez-Ballester gravity [7, 8]; Gauss-Bonnet gravity and its different generalizations [9-11], and the well-known teleparallel theory and its extended versions [12-16] like f(T) theory, f(T, B) gravity and $f(T,{ \mathcal T })$ theory, where T, B and ${ \mathcal T }$ refer to the torsion scalar, divergence term and the energy-momentum tensor trace, respectively. In this respect, another significant framework was proposed by Harko et al [17] where they generalized the Lagrangian of the f(R) theory by involving a coupling between the matter and curvature; this is termed the ${\mathbb{F}}(R,{ \mathcal T })$ theory. This theory has been widely tested on cosmological landscape and has been found to be flawless in explaining various cosmological aspects [18-23]. Its modified versions, namely $f(R,{{\mathbb{L}}}_{m})$ gravity and the $f(R,{{\mathbb{L}}}_{m},{ \mathcal T })$ theory, have also been proposed in the literature and numerous researchers have explored their significance on astrophysical grounds [24, 25].
The search for a viable wormhole geometry in the context of GR as well as MGFs has been revealed as one of the most worthwhile lines of action for researchers during last few decades. A tunnel-like geometrical structure with two open mouths which are linked through a throat is known as a wormhole. It is argued that the existence of such tunnels can offer fast humanoid travel options between different regions of the universe or different universes. In the literature, a considerable amount of work has been done on this topic in GR as well as other gravity theories and interesting outcomes have been achieved. Initially, Flamm [26] worked on this idea and put forward a model representing an interstellar bridge by taking the Schwarzschild black hole into account. Further, another interesting model, namely the 'Einstein-Rosen bridge', was presented mathematically by Einstein and Rosen [27]. For such objects, the 'wormhole' label was first used by Misner and Wheeler [28]. Morris and Thorne [29] proposed the idea of traversable wormholes and argued that the development and stability of such structures demands the presence of an unusual sort of matter, i.e. exotic matter (a type of matter that disobeys the well-known null-energy condition (NEC)). At the classical level, the accessibility of such matter is a major issue for researchers, however, quantum mechanics can play a vital role in this respect as the vacuum fluctuations of quantum fields can generate negative energy. Recently, this idea has gained much popularity and numerous researchers have utilized the energy density of the renowned Casimir's effect (1948). It is argued [30] that a force may exist between two parallel-placed conducting plates without charge and that it is due to the disturbances resulting from the electromagnetic field vacuum. Basically, this idea was connected with the concept of zero point energy put forward by Niel Bohar, which was confirmed experimentally later [31]. For a setup of two parallel-placed plates, the pressure and density (due to the Casimir effect) can be, respectively, defined as
$\begin{eqnarray}p({\rm{a}})=-\frac{{\pi }^{2}}{240}\times \left(\frac{1}{{{\rm{a}}}^{4}}\right),\quad \rho ({\rm{a}})=-\frac{{\pi }^{2}}{720{{\rm{a}}}^{4}}.\end{eqnarray}$
Herein, a represents the distance between the plates. Also, it is easy to check that p = ωρ with ω = 3.Several wormhole models exhibiting a spherically-symmetric nature have been formulated in the literature and their viability has been examined graphically in different gravitational frameworks [32-40]. Here, we shall give a quick synopsis of some captivating and dominant works which are present in the literature. Mainly, the available works can be grouped into two categories. In the first, a specific form of energy density is assumed and the resulting shape model is calculated and examined for its essential features. In the second category, an ansatz for a wormhole shape model is considered and the validity of the energy constraint is then tested for fixing the free model parameters appropriately. Another more general way is to use the well-known Noether symmetries along with conserved quantities for computing the wormhole geometry by fixing unknowns [41]. In ${\mathbb{F}}(R,{ \mathcal T })$ gravity, Azizi [42] found the wormhole shape model mathematically which was compatible with the NEC bound. Zubair et al [43] explored the wormhole geometries by utilizing three sorts of matter contents (isotropic, barotropic and anisotropic fluids) as well as the Starobinsky model as an option for the f1(R) function where they assumed ${\mathbb{F}}(R,{ \mathcal T })={f}_{1}(R)+{f}_{2}({ \mathcal T })$. With the help of energy constraints, they provided the possible constraints on free model parameters. Elizalde and Khurshudyan [44] proposed a general wormhole construction by linking the ordinary matter energy density with the Ricci scalar in the ${\mathbb{F}}(R,{ \mathcal T })$ gravitational framework, where they considered linear as well as quadratic models for the ${\mathbb{F}}(R,{ \mathcal T })$ function and obtained worthy results.
In another paper, Chanda et al [45] discussed the wormhole construction in ${\mathbb{F}}(R,{ \mathcal T })$ gravity by assuming ${\mathbb{F}}(R,{ \mathcal T })=R+\alpha {R}^{2}+\lambda {{ \mathcal T }}^{\beta }$ with anisotropic matter; they concluded that the proposed wormhole geometry is highly dependent on parameters λ and α. In ${\mathbb{F}}(R,{ \mathcal T })$ gravity, Godani [46] used the power-law form of the red-shift function and the logarithmic form of the shape model to discuss wormhole geometry and examined the compatibility of corresponding energy constraints. Mondal and Rahaman [47] explored galactic wormholes in linear ${\mathbb{F}}(R,{ \mathcal T })$ theory using some interesting density profiles. Moraes et al [48] explored charged wormholes in the context of the ${\mathbb{F}}(R,{ \mathcal T })$ framework and showed that the involvement of charge in such structures enhances the stability and also that the energy constraints are valid in such cases. Banerjee et al [49] studied spherically-symmetric wormholes with isotropic pressure to check the possibility of getting wormholes without the involvement of any exotic matter. Tangphati et al [50] investigated wormholes in ${\mathbb{F}}(R,{ \mathcal T })$ theory by taking two options, Lm = ρ and Lm = pr, and discussed the respective outcomes. Mustafa et al [51] considered the exponential and Tsujikawa models for the ${\mathbb{F}}(R,{ \mathcal T })$ function and explored the respective wormhole shape models. Godani and Samanta [52] utilized the radial and tangential equation of state (EoS) parameters along with a non-linear f(R, T) function and presented the resulting wormhole solutions. Moraes et al [53] discussed wormholes in a novel theory, namely ${\mathbb{F}}(R,L,{ \mathcal T })$ gravity, by assuming the simplest choice of linear function in these quantities and checked the respective energy constraints graphically.
On the construction of wormhole geometry using the Casimir source, Garattini [54-56] has done much work and used the 'Casimir wormhole' title for the first time. He also involved the Yukawa-correction term in Casimir density to obtain the wormhole model and discussed the resulting solutions. In the framework of f(R) theory, Sokoliuk et al [57] studied Casimir wormholes by taking the zero tidal force condition along with the quadratic as well as power-law forms of the f(R) function and examined the viability of the obtained solutions. Tripathy [58] utilized the density of Casimir's effect with Generalized Uncertainty Principal (GUP) corrections to find the wormhole shape model in the ${\mathbb{F}}(R,{ \mathcal T })$ framework and discussed its main properties graphically. Alencar et al [59] discussed the construction of a wormhole in (2 + 1) dimensions by involving the Casimir source. Carvalho et al [60] computed the gravitational deflection angle of light and massive particles due to Casimir wormholes and found interesting results. Avalos et al [61] put forward some wormhole models which are based on the idea of a non-vanishing complexity factor and explored the significance of their proposed solutions. Banerjee et al [62] found some interesting solutions by involving the Casimir effect in ${\mathbb{F}}(R,{ \mathcal T })$ theory. They picked different options for wormhole red shift and discussed the resulting models graphically.
Azmat et al [63] investigated the impact of electric charge on Casimir wormholes in ${\mathbb{F}}(R,{ \mathcal T })$ gravity where they considered three interesting setups of the Casimir effect along with its GUP corrected form and obtained worthy outcomes. Zubair and Farooq [64] formulated some wormhole models in Einstein Gauss-Bonnet (EGB) theory by involving Casimir density along with its GUP corrected form and examined the role of the Gauss-Bonnet (GB) coupling parameter in this regard. In a recent study, Kuhfittig [65] attempted to relate the Casimir effect with the well-known non-commutative geometry. The idea of conformal killing vectors (CKVs) along with non-commutative distributions have been utilized to compute the wormhole shape model in ${\mathbb{F}}(R,{ \mathcal T })$ gravity [66] and the resulting solutions were tested by exploring primary wormhole properties graphically. Banerjee et al [67] constructed some viable wormhole models by involving CKVs in ${\mathbb{F}}(R,{ \mathcal T })$ gravity. They explored the existence of wormholes by taking isotropic pressure, then anisotropic fluid with radial and tangential EoS parameters and lastly utilized a relation, pt = npr, as separate cases. In [68], the authors utilized CKVs along with linear EoS and traceless fluid conditions to explore the form of the wormhole shape function in f(Q) theory.
In symmetric teleparallel theories, namely the f(Q) and f(Q, T) theories, the Casimir effect has been widely used to explore the form of a wormhole shape model and numerous interesting and viable works are available [69-71]. Oliveira et al [72] used three models for the wormhole shape function with Yukawa correction to study the nature and stability of wormhole geometry; they found that wormholes with repulsive gravitational nature can be generated for some Yukawa parameter values. Jawad et al [73] used the Yukawa-corrected form of the density profile in torsion-matter coupled theory and explored the nature and stability of the resulting wormhole solutions. Recently, Kumar et al [74] have considered the Yukawa-corrected wormhole shape model to discuss wormhole geometry in ${\mathbb{F}}(R,{ \mathcal T })$ theory and investigated the impact of the Yukawa correction parameter on the formation and stability of wormholes.
Kumar et al [75] presented a comprehensive overview of wormhole construction in different gravitational frameworks like f(R), f(G), f(T), f(R, Lm), f(R, T), f(Q), and f(Q, T) theories. Further, the stability analysis of the discussed solutions has been carried through energy conditions, the TOV equation, VIQ, active gravitational mass, and total gravitational energy. Kiroriwal et al [76] investigated the existence of wormhole models in f(Q) theory where the non-linear function of f(Q) gravity is taken into account. They checked the validity of energy conditions and discussed the respective embedding diagram as well as the volume integral quantifier. Chaudhary et al [77] studied the wormhole solutions in non-minimally coupled F(R, Lm) gravity by considering two newly-proposed shape functions. It was seen that the NEC shows negative behavior; further, they discussed the VIQ and the TOV equation to demonstrate the physical viability of their obtained solutions. Next, Kiroriwal et al [78] explored the existence and stability of wormholes in well-known F(R, Lm) gravity by taking the linear F(R, Lm) form along with two shape functions. They demonstrated the resulting wormhole geometry through embedding the diagram in two and three-dimensional Euclidean spaces and examined the stability of solutions through the TOV equation. Further, they discussed the complexity factor for both models graphically. In another study [79], Chaudhary et al studied wormholes in F(R, T) theory where they used F(R, T) = ξR2 + R + ηT along with two specific and interesting forms of shape function. They analyzed the stability and physical validity of the used shape functions through different parameters and obtained interesting outcomes. In another work, Maurya et al [80] studied wormhole geometry by taking two interesting dark matter density profiles, namely the Einasto spike and the pseudo-isothermal models in f(R, Lm) theory. Through graphical analysis of different measures like the adiabatic index and anisotropy ratio, they concluded that their obtained wormhole solutions are physically promising. Further, Kiroriwal et al [81] discussed a wormhole model exhibiting asymptotically flat behavior in f(Q) theory by taking a specific EoS parameter. In this respect, they have developed two wormhole models and explored the impact of the EoS parameter on these models. Chaudhary et al [82] studied the properties of wormhole models in Rastall gravity by considering two types of EoS parameter and concluded that both formulated models are physically feasible. In another study, Kumar et al [83] explored the properties of wormholes in f(Q, T) theory where f(Q, T) = σQ + λT is considered. By taking two dark matter profiles, namely the Einasto spike and pseudo isothermal, they developed the respective wormhole solutions and check their physical validity through different measures.
The present article is an interesting extension of previously-discussed works as it involves the Yukawa-correction term in Casimir energy density. To accomplish this task, we shall consider spherical symmetric wormhole geometry with anisotropic fluid and a linear form of the ${\mathbb{F}}(R,{ \mathcal T })$ function. We first take simple ${\mathbb{F}}(R,{ \mathcal T })$ field equations and introduce the Yukawa-corrected Casimir energy density to obtain the analytical form of the wormhole shape function. Secondly, we solve the ${\mathbb{F}}(R,{ \mathcal T })$ field equations after introducing the CKVs and find the exact wormhole shape function by taking Yukawa-corrected Casimir density. Thirdly, we consider a recently-proposed shape function and discuss the validity of energy constraints. We also explore the viability of all shape models by investigating the behavior of different measures graphically. The present work is organized as follows. The upcoming section provides a short introduction to ${\mathbb{F}}(R,{ \mathcal T })$ gravity along with some necessary assumptions used to accomplish this work. In section 3 , we find the analytical form of the shape function using Yukawa-corrected Casimir energy with or without involving CKVs and discuss the properties of the obtained functions. In the same segments, we provide the corresponding graphical analysis of various quantities. We also analyze a newly-proposed wormhole shape model in ${\mathbb{F}}(R,{ \mathcal T })$ theory with CKVs. A summary of the whole study is provided in the last segment.
2. ${\mathbb{F}}(R,{ \mathcal T })$ dynamical field equations and some primary assumptions
This segment briefly reviews the background formulation as well as the dynamical equations of well-known ${\mathbb{F}}(R,{ \mathcal T })$ gravity. We demonstrate the primary criteria for a valid spherically-symmetric wormhole geometry along with some essential assumptions. Action for ${\mathbb{F}}(R,{ \mathcal T })$ theory of gravity [17] is defined as
$\begin{eqnarray}{ \mathcal A }=\frac{1}{16\pi }\int {\mathbb{F}}(R,{ \mathcal T })\sqrt{-g}\,{{\rm{d}}}^{4}x+\int {{\mathbb{L}}}_{m}\sqrt{-g}\,{{\rm{d}}}^{4}x,\end{eqnarray}$
where the Lagrangian function ${\mathbb{F}}(R,{ \mathcal T })$ is based on the Ricci scalar R and the trace of energy-momentum tensor (${ \mathcal T }={g}^{\mu \nu }{{ \mathcal T }}_{\mu \nu }$). Here, the ordinary matter Lagrangian density is denoted by ${{\mathbb{L}}}_{m}$ and is defined by $\begin{eqnarray}{{ \mathcal T }}_{\mu \nu }=-\frac{2}{\sqrt{-g}}\frac{\delta (\sqrt{-g}{{\mathbb{L}}}_{m})}{\delta {g}^{\mu \nu }}.\end{eqnarray}$
Adopting the same strategy used by Harko et al [17], we suppose that the Lagrangian density is dependent on gμν only and consequently, the following outcome can be deduced: $\begin{eqnarray}{{ \mathcal T }}_{\mu \nu }={g}_{\mu \nu }{{\mathbb{L}}}_{m}-2\frac{\partial {{\mathbb{L}}}_{m}}{\partial {g}^{\mu \nu }}.\end{eqnarray}$
The metric tensor variation of action ${ \mathcal A }$ results in the following set of field equations: $\begin{eqnarray}\begin{array}{rcl} & & 8\pi {{ \mathcal T }}_{\mu \nu }-{{\mathbb{F}}}_{T}(R,{ \mathcal T }){{ \mathcal T }}_{\mu \nu }-{{\mathbb{F}}}_{{ \mathcal T }}(R,{ \mathcal T }){{\rm{\Xi }}}_{\mu \nu }\\ & & \quad ={{\mathbb{F}}}_{R}(R,{ \mathcal T }){R}_{\mu \nu }-\frac{1}{2}{\mathbb{F}}(R,{ \mathcal T }){g}_{\mu \nu }+({g}_{\mu \nu }\square -{{\rm{\nabla }}}_{\mu }{{\rm{\nabla }}}_{\nu }){{\mathbb{F}}}_{R}(R,{ \mathcal T }).\end{array}\end{eqnarray}$
Herein, the introduced symbols ∇ and □ refer to the covariant derivative and box or d'Alembert operator, respectively. Further, the first-order time rates of the generic function with respect to R and ${ \mathcal T }$ are denoted by ${{\mathbb{F}}}_{R}(R,{ \mathcal T })$ and ${{\mathbb{F}}}_{{ \mathcal T }}(R,{ \mathcal T })$, respectively. Also, Ξμν is given by $\begin{eqnarray}{{\rm{\Xi }}}_{\mu \nu }=\frac{{g}^{ij}\delta {T}_{ij}}{\delta {g}^{\mu \nu }}=-2{T}_{\mu \nu }+{g}_{\mu \nu }{{\mathbb{L}}}_{m}-2{g}^{ij}\frac{{\partial }^{2}{{\mathbb{L}}}_{m}}{\partial {g}^{ij}\partial {g}^{\mu \nu }}.\end{eqnarray}$
For ordinary matter contents, let us assume the energy-momentum tensor of the anisotropic fluid source, which is defined as $\begin{eqnarray}{{ \mathcal T }}_{\mu \nu }=(\rho +{p}_{t}){v}_{\mu }{v}_{\nu }-{p}_{t}{g}_{\mu \nu }+({p}_{r}-{p}_{t}){u}_{\mu }{u}_{\nu }.\end{eqnarray}$
Here, the notations ρ, pr and pt stand for the energy density and pressure components in radial and tangential directions, respectively, while the 4-vectors uμ and vν are compatible with relations vμvν = 1 and uμuν = − 1. For the present study, we consider the static wormhole geometry exhibiting a spherically-symmetric nature, which is defined by the following metric: $\begin{eqnarray}{\rm{d}}{s}^{2}={{\rm{e}}}^{\alpha (r)}{\rm{d}}{t}^{2}-{{\rm{e}}}^{\beta (r)}{\rm{d}}{r}^{2}-{r}^{2}({\rm{d}}{\theta }^{2}+{\rm{s}}{\rm{i}}{{\rm{n}}}^{2}\theta {\rm{d}}{\phi }^{2}),\end{eqnarray}$
where the unknown radial-coordinate dependent functions α(r) and β(r) can be further, respectively, linked to the red-shift ϵ(r) and shape S(r) functions of the wormhole through the following relations: $\begin{eqnarray}\alpha (r)=2\varepsilon (r),\quad \beta (r)=-\mathrm{ln}(1-\frac{S}{r}).\end{eqnarray}$
For the present study, we suppose that ${{\mathbb{L}}}_{m}=-{\mathbb{P}}=-\frac{{p}_{r}+2{p}_{t}}{3}$ and then equation (6 ) turns out as follows
$\begin{eqnarray}{{\rm{\Xi }}}_{\mu \nu }=-2{{ \mathcal T }}_{\mu \nu }-{\mathbb{P}}{g}_{\mu \nu }.\end{eqnarray}$
In order to minimize the complexity of the dynamical equations, we use a simple linear form of the Lagrangian function ${\mathbb{F}}(R,{ \mathcal T })$; it is given by ${\mathbb{F}}(R,{ \mathcal T })=R+2\zeta { \mathcal T }$, where ζ corresponds to the coupling constant. It is easy to see that the ${\mathbb{F}}(R)$ theory can be recovered when the coupling constant vanishes. By making use of all these assumptions, the set of dynamical equations can be solved to isolate the state variables ρ, pr and pt and the resulting expressions can be written as [63, 84] $\begin{eqnarray}\rho =\frac{S^{\prime} (r)(\zeta +3\pi )}{3{r}^{2}(\zeta +2\pi )(\zeta +4\pi )},\end{eqnarray}$
$\begin{eqnarray}{p}_{r}=\frac{\zeta rS^{\prime} (r)-3(\zeta +2\pi )S(r)}{6{r}^{3}(\zeta +2\pi )(\zeta +4\pi )},\end{eqnarray}$
$\begin{eqnarray}{p}_{t}=\frac{3S(r)(\zeta +2\pi )-S^{\prime} (r)r(\zeta +6\pi )}{12{r}^{3}(\zeta +2\pi )(\zeta +4\pi )}.\end{eqnarray}$
Herein, one needs to impose ζ ≠ − 2π, − 3π, − 4π. Also, it is worth mentioning here that we have utilized the zero tidal force condition, i.e. ϵ(r) = 0.
The Yukawa-correction term has been introduced in the literature to discuss wormhole solutions in two ways: either modifying the wormhole shape function due to this term or by taking the Yukawa-corrected form of matter density. Firstly, Garattini [56] introduced this Yukawa-correction term to modify the shape function of a Casimir wormhole (by multiplying $f(r)={{\rm{e}}}^{-\mu (r-{r}_{0})}$). It is pointed out that on physical grounds, such modification is linked to the existence of massive modes with negative energy. Yukawa [85-89] put forward a version of the Coulomb potential that explains the non-relativistic strong interactions between nucleons and is given by $V(r)=-\frac{\chi }{r}{{\rm{e}}}^{\alpha r}$. Herein, χ refers to the strength of nucleons' interaction and its range is given by $\frac{1}{\alpha }$. Numerous researchers utilized this short range interaction to comprehend the deviations of the Newtonian potential. Consequently, it is argued that the gravitational Newtonian potential may also be modified by including the Yukawa correction as $V(r)=-\frac{G{m}_{1}{m}_{2}}{r}(1+\mu {{\rm{e}}}^{\alpha r}),$ where m1 and m2 correspond to two atomic masses placed at distance r from each other. It is worth pointing out here that many astrophysicists examined such potentials to study graviton masses. In the literature, it is mentioned that such Yukawa-type forces also appeared in modified gravity theories as well as bigravity theories. A Yukawa term seems to be directly involved in the Galaxy rotation curves [90] and further, a relation between the Casimir forces and the Yukawa profile has also been presented in [91]. Since the Casimir density, being negative energy, can provide an artificial way to create exotic material which is one of the key requirements in wormhole construction, therefore the Yukawa-corrected form of Casimir density can also lead to worthy outcomes. After Garattini, many other researchers have used some shape function or density anzatz with the Yukawa-correction term in the context of different modified gravity theories and obtained interesting outcomes [92, 93].
In this work, we shall use the energy density of the Casimir effect with the Yukawa-correction. It is defined as [73, 94]
$\begin{eqnarray}\rho =-\frac{{\pi }^{2}}{2r(720{r}^{4})}\left(\gamma r+\sigma {r}_{0}{{\rm{e}}}^{-\eta (r-{r}_{0})}\right).\end{eqnarray}$
Herein, σ, r0, γ and η > 0 all are arbitrary constants. It is well-known that for a viable wormhole geometry, the wormhole red-shift and shape functions must meet the following requirements:In this study, we shall first examine the proposed wormholes by investigating the validity of energy constraints. Their stability will be then explored through the adiabatic index and the Tolman-Oppenheimer-Volkov (TOV) equation. We shall also compute some important measures, namely the volume integral quantifier (VIQ), the complexity factor, the exoticity factor and the active mass energy density for all cases. The energy constraints are basically some inequalities in terms of density and pressure components and demonstrate the physical viability of any gravitational configuration. For the present scenario, the inequalities defining the null (NEC), the weak (WEC), the strong (SEC) and the dominant energy (DEC) bounds are given by
$\begin{eqnarray}\begin{array}{rcl} & & NEC:\,\rho +{p}_{r}\geqslant 0,\,\,\rho +{p}_{t}\geqslant 0,\\ & & WEC:\,\rho \geqslant 0,\,\rho +{p}_{r}\geqslant 0,\,\rho +{p}_{t}\geqslant 0,\quad SEC:\,\rho -2{p}_{r}+{p}_{t}\geqslant 0,\\ & & DEC:\,\rho -| {p}_{r}| ,\,\rho -| {p}_{t}| \geqslant 0.\end{array}\end{eqnarray}$
In order to explore the stability, we shall firstly check the behavior of forces present in the well-known TOV equation [95]. The TOV equation is defined in terms of three forces, namely gravitational force Fg, the hydrostatic force Fh and the anisotropic force Fa and indicates that the sum of all three forces must vanish to achieve the stability of any system, i.e. $\begin{eqnarray}\begin{array}{rcl} & & {F}_{g}+{F}_{h}+{F}_{a}=0;\quad {F}_{g}={\rm{\Phi }}^{\prime} (r)\left(\right.\rho +{p}_{r}\left)\right.,\quad {F}_{h}=-\frac{{\rm{d}}{p}_{r}}{{\rm{d}}r},\\ & & {F}_{a}=2\left(\right.\frac{{p}_{r}-{p}_{t}}{2}\left)\right..\end{array}\end{eqnarray}$
An additional measure to examine the stability of any configuration is the adiabatic index which is defined as $\begin{eqnarray}{\rm{\Gamma }}=\frac{\rho +{p}_{r}}{{p}_{r}}\frac{{\rm{d}}{p}_{r}}{{\rm{d}}\rho }.\end{eqnarray}$
It is argued that for a stable configuration, the corresponding adiabatic index must exceed the threshold value $\frac{4}{3}$ for all r. Within a region starting from a wormhole throat r1 to a radius R, the active gravitational mass is denoted by ${{\mathbb{M}}}_{A}$ and can be defined as [63] $\begin{eqnarray}{{\mathbb{M}}}_{A}=4\pi {\int }_{{r}_{1}}^{R}\rho (r){r}^{2}{\rm{d}}r.\end{eqnarray}$
It is worth pointing out here that for a physically valid model, the active gravitational mass must exhibit positive conduct while for exotic matter, it can take negative values. Further, another important measure in this respect is the VIQ, an integral specifying the amount of exotic matter required to guarantee the wormhole existence; it is given by $\begin{eqnarray}VIQ=\oint (\rho +{p}_{r}){\rm{d}}v=8\pi {\int }_{{r}_{1}}^{{r}_{2}}(\rho +{p}_{r}){r}^{2}{\rm{d}}r,\end{eqnarray}$
where r2 represents the introduced cutoff and corresponds to the extension of wormhole geometry from the wormhole throat r1 to a value r2 where r2 > r1. It is seen that the VIQ exhibits negative behavior due to the presence of exotic material and it must approach zero when r2 → r1.The exoticity factor is another interesting measure which helps us to investigate the presence of exotic materials at the wormhole's neck as well as its neighborhood; it is defined as
$\begin{eqnarray}\Upsilon =-\frac{\rho -{p}_{r}}{| \rho | }.\end{eqnarray}$
It is argued that the non-negative behavior of this parameter indicates the presence of exotic matter at the wormhole's neck and its surroundings, while the negative behavior of this factor refers to the presence of exotic matter only near the wormhole's neck with normal matter far from the throat. In this respect, Herrera [96] put forward a significant quantity, namely the complexity factor for a self gravitating source exhibiting a spherically-symmetric nature. It is argued that a minimal amount of complexity can be achieved for a homogeneous fluid involving isotropic pressure (complexity factor vanishes in this case). This scalar appears from the orthogonal splitting of the Riemann tensor and provides general information about the complexity of a system. For anisotropic matter, the complexity factor can be defined by the below integral: $\begin{eqnarray}{Y}_{TF}=8\pi ({p}_{r}-{p}_{t})-\frac{1}{2{r}^{3}}{\int }_{{r}_{1}}^{r}{r}^{3}\rho ^{\prime} (r){\rm{d}}r.\end{eqnarray}$
It is observed that the value of this integral must approach zero when r → ∞ [63].3. Yukawa-Casimir wormholes with or without CKVs in ${\mathbb{F}}(R,{ \mathcal T })$ gravity
Here we shall construct the possible analytical forms of the wormhole shape function by taking the Yukawa-corrected Casimir energy in ${\mathbb{F}}(R,{ \mathcal T })$ theory where the conformal symmetries are involved and discuss their properties graphically. In upcoming segments, we shall explore wormhole geometry for the below cases:
3.1. Yukawa-corrected Casimir wormholes in ${\mathbb{F}}(R,{ \mathcal T })$ theory
In this part, we shall determine the analytical solution of unknowns of wormhole geometry by utilizing the Yukawa-corrected form of Casimir energy (between two parallel-placed plates). We shall also explore the behavior of obtained functions through graphical analysis of different quantities. For the wormhole metric (8 ), the energy density (where we have considered: ϵ(r) = 0) is given by 14 ) and (22 ) as follows
$\begin{eqnarray}\rho =\frac{S^{\prime} (r)(\zeta +3\pi )}{3{r}^{2}(\zeta +2\pi )(\zeta +4\pi )}.\end{eqnarray}$
To find the wormhole shape model, we equate the energy densities given by equations ( $\begin{eqnarray*}\frac{S^{\prime} (r)(\zeta +3\pi )}{3{r}^{2}(\zeta +2\pi )(\zeta +4\pi )}=-\frac{{\pi }^{2}}{2r(720{r}^{4})}\left(\gamma r+\sigma {r}_{0}{{\rm{e}}}^{-\eta (r-{r}_{0})}\right).\end{eqnarray*}$
The solution of the resulting Ordinary Differential Equation (ODE) is given by $\begin{eqnarray*}\begin{array}{rcl}S(r) & = & {{ \mathcal C }}_{1}-\frac{1}{960{r}^{2}(3\pi +\zeta )}\\ & & \times \,\left[{{\rm{e}}}^{-r\eta }{\pi }^{2}(8{\pi }^{2}+6\pi \zeta +{\zeta }^{2})\left(-2{{\rm{e}}}^{r\eta }r\gamma -{{\rm{e}}}^{{r}_{0}\eta }{r}_{0}(-1+r\eta )\sigma \right.\right.\\ & & +\,\left.\left.{{\rm{e}}}^{(r+{r}_{0})\eta }{r}^{2}{r}_{0}{\eta }^{2}\sigma \mathrm{ExpIntegral}E{\rm{i}}[-r\eta ]\right)\right].\end{array}\end{eqnarray*}$
Herein, ExpIntegralEi[z] denotes the exponential integral function with $E{\rm{i}}[z]=-{\int }_{-z}^{\infty }\frac{{{\rm{e}}}^{-t}}{t}{\rm{d}}t$. The integration constant ${{ \mathcal C }}_{1}$ can be fixed by utilizing the condition S(r1) = r1 and is given by $\begin{eqnarray*}\begin{array}{rcl}{{ \mathcal C }}_{1} & = & {r}_{1}+\frac{1}{960{{r}_{1}}^{2}(3\pi +\zeta )}\left[{{\rm{e}}}^{-{r}_{1}\eta }{\pi }^{2}(8{\pi }^{2}+6\pi \zeta +{\zeta }^{2})\right.\\ & & \times \,\left(-2{{\rm{e}}}^{{r}_{1}\eta }{r}_{1}\gamma +{{\rm{e}}}^{{r}_{0}\eta }{r}_{0}(-1+{r}_{1}\eta )\sigma \right.\\ & & +\,\left.\left.{{\rm{e}}}^{({r}_{0}+{r}_{1})\eta }{r}_{0}{{r}_{1}}^{2}{\eta }^{2}\sigma \mathrm{ExpIntegral}E{\rm{i}}[-{r}_{1}\eta ]\right)\right].\end{array}\end{eqnarray*}$
Consequently, the wormhole shape function takes the following form: $\begin{eqnarray}\begin{array}{rcl}S(r) & = & {r}_{1}-\frac{1}{960(\zeta +3\pi )}\left[\frac{1}{{r}^{2}}\left({{\rm{e}}}^{-r\eta }{\pi }^{2}(8{\pi }^{2}+6\pi \zeta +{\zeta }^{2})\right.\right.\\ & & \times \,(-2{{\rm{e}}}^{r\eta })r\gamma +{{\rm{e}}}^{{r}_{0}\eta }{r}_{0}(-1-r\eta )\sigma \\ & & +\,\left.{{\rm{e}}}^{(r+{r}_{0})\eta }{r}^{2}{r}_{0}{\eta }^{2}\sigma \mathrm{ExpIntegral}E{\rm{i}}[-r\eta ]\right)\\ & & -\,\frac{1}{{{r}_{1}}^{2}}\left({{\rm{e}}}^{-{r}_{1}\eta }{\pi }^{2}(8{\pi }^{2}+6\pi \zeta +{\zeta }^{2})(-2{{\rm{e}}}^{{r}_{1}\eta }{r}_{1}\gamma \right.\\ & & +\,{{\rm{e}}}^{{r}_{0}\eta }){r}_{0}(-1+{r}_{1}\eta )\sigma \\ & & +\,\left.\left.{{\rm{e}}}^{({r}_{0}+{r}_{1})\eta }{r}_{0}{{r}_{1}}^{2}{\eta }^{2}\sigma \mathrm{ExpIntegral}E{\rm{i}}[-{r}_{1}\eta ]\right)\right].\end{array}\end{eqnarray}$
By utilizing the above value of S(r), the pressure components in the radial and tangential directions become $\begin{eqnarray}\begin{array}{rcl}{p}_{r} & = & -\frac{1}{5760{r}^{5}{{r}_{1}}^{2}(2\pi +\zeta )(3\pi +\zeta )(4\pi +\zeta )}\\ & & \times \,\left[2{{\rm{e}}}^{-r\eta }{\pi }^{2}{{r}_{1}}^{2}\zeta (8{\pi }^{2}+6\pi \zeta +{\zeta }^{2})({{\rm{e}}}^{r\eta }r\gamma \right.\\ & & +\,{{\rm{e}}}^{{r}_{0}\eta }{r}_{0}\sigma )+3(2\pi +\zeta )\\ & & \times \,\left(960{r}^{2}{{r}_{1}}^{3}(3\pi +\zeta )+{{\rm{e}}}^{-r\eta }{\pi }^{2}{{r}_{1}}^{2}(8{\pi }^{2}+6\pi \zeta +{\zeta }^{2})\right.\\ & & (2{{\rm{e}}}^{r\eta }r\gamma -{{\rm{e}}}^{{r}_{0}\eta }{r}_{0}(-1+r\eta )\sigma -{{\rm{e}}}^{(r+{r}_{0})\eta }{r}^{2}{r}_{0}{\eta }^{2}\sigma \\ & & \times \,\mathrm{ExpIntegral}E{\rm{i}}[-r\eta ])-{{\rm{e}}}^{-{r}_{1}\eta }{\pi }^{2}{r}^{2}(8{\pi }^{2}+6\pi \zeta +{\zeta }^{2})\\ & & \times \,(2{{\rm{e}}}^{{r}_{1}\eta }{r}_{1}\gamma -{{\rm{e}}}^{{r}_{0}\eta }{r}_{0}(-1+{r}_{1}\eta )\sigma -{{\rm{e}}}^{({r}_{0}+{r}_{1})\eta }{r}_{0}{{r}_{1}}^{2}{\eta }^{2}\sigma \\ & & \times \,\left.\left.\mathrm{ExpIntegral}E{\rm{i}}[-{r}_{1}\eta ]\right)\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{p}_{t} & = & \frac{1}{11520{r}^{5}{{r}_{1}}^{2}(2\pi +\zeta )(3\pi +\zeta )(4\pi +\zeta )}\\ & & \times \,\left[2{{\rm{e}}}^{-r\eta }{\pi }^{2}{{r}_{1}}^{2}(\zeta +6\pi )(8{\pi }^{2}+6\pi \zeta +{\zeta }^{2})({{\rm{e}}}^{r\eta }r\gamma +{{\rm{e}}}^{{r}_{0}\eta }{r}_{0}\sigma )\right.\\ & & +\,3(2\pi +\zeta )\left(960{r}^{2}{{r}_{1}}^{3}(3\pi +\zeta )+{{\rm{e}}}^{-r\eta }{\pi }^{2}{{r}_{1}}^{2}(8{\pi }^{2}+6\pi \zeta +{\zeta }^{2})\right.\\ & & \times \,(2{{\rm{e}}}^{r\eta }r\gamma -{{\rm{e}}}^{{r}_{0}\eta }{r}_{0}(-1+r\eta )\sigma \\ & & -\,{{\rm{e}}}^{(r+{r}_{0})\eta }{r}^{2}{r}_{0}{\eta }^{2}\sigma \mathrm{ExpIntegral}E{\rm{i}}[-r\eta ])\\ & & -\,{{\rm{e}}}^{-{r}_{1}\eta }{\pi }^{2}{r}^{2}(8{\pi }^{2}+6\pi \zeta +{\zeta }^{2})\\ & & \times \,(2{{\rm{e}}}^{{r}_{1}\eta }{r}_{1}\gamma -{{\rm{e}}}^{{r}_{0}\eta }{r}_{0}(-1+{r}_{1}\eta )\sigma \\ & & -\,\left.\left.{{\rm{e}}}^{({r}_{0}+{r}_{1})\eta }{r}_{0}{{r}_{1}}^{2}{\eta }^{2}\sigma \mathrm{ExpIntegral}E{\rm{i}}[-{r}_{1}\eta ])\right)\right].\end{array}\end{eqnarray}$
The basic properties of the shape model S(r) have been examined in the graphs of figures 1 and 2. It can be seen that the function S(r) exhibits positive and decreasing behavior versus r and $S^{\prime} (r)$ which is in accordance with the inequality $S^{\prime} ({r}_{1})\lt 1$; however, it indicates negative behavior, as expected (for a decreasing wormhole shape model). Further, plots of figure 2 show that for the calculated S(r) function, the asymptotic flatness property is not achieved ($1-\frac{S(r)}{r}\approx 0.8$) while S(r) − r < 0 holds. In addition, the validity of NEC inequalities is tested in the plots of figure 3. It can be easily checked that the NEC for radial pressure is violated for all ζ values while the second inequality for the tangential pressure component remains valid. It is worth pointing out here that for the majority of ζ choices, both these inequalities are not satisfied near the throat value. Collectively, it can be inferred that the NEC remains invalid and hence ensures the presence of exotic matter.
Figure 1. Left and right plots, respectively, correspond to the behavior of the wormhole shape model and its rate of change with respect to r. Here we use: r0 = 2, γ = 2.5, σ = − 2, η = 1, and r1 = 2. |
Figure 2. Left and right graphs represent the behavior of conditions $1-\frac{S(r)}{r}$ and S(r) − r, respectively. |
Figure 3. Plots refer to the behavior of the inequalities of NEC versus ζ and r values. |
Further, the EoS parameters for radial and tangential pressures can be found as follows 28 ), hence ensuring the presence of exotic matter. In the right plot of figure 5, the behavior of the anisotropy parameter is provided, which indicates the negative behavior for all ζ choices and hence refers to the attractive nature of gravity within a wormhole. Further, we compute the VIQ for the proposed wormhole shape model. In the present case, equation (19 ) results in the following integral: 16 ). In the present case, these forces take the following form:
$\begin{eqnarray}\begin{array}{rcl}{\omega }_{r}(r) & = & \frac{1}{4{\pi }^{2}{{r}_{1}}^{2}(2\pi +\zeta )(3\pi +\zeta )(4\pi +\zeta )(r\gamma +{{\rm{e}}}^{(-r+{r}_{0})\eta }{r}_{0}\sigma )}\\ & & \times \,\left[2{{\rm{e}}}^{-r\eta }{\pi }^{2}{{r}_{1}}^{2}\zeta (8{\pi }^{2}+6\pi \zeta +{\zeta }^{2})({{\rm{e}}}^{r\eta }r\gamma +{{\rm{e}}}^{{r}_{0}\eta })\right.\\ & & \times \,{r}_{0}\sigma +3(2\pi +\zeta )\left(960{r}^{2}{{r}_{1}}^{3}(3\pi +\zeta )\right.\\ & & +\,{{\rm{e}}}^{-r\eta }{\pi }^{2}{{r}_{1}}^{2}(8{\pi }^{2}+6\pi \zeta +{\zeta }^{2})(2{{\rm{e}}}^{r\eta }r\gamma -{{\rm{e}}}^{{r}_{0}\eta }{r}_{0}(-1+r\eta )\sigma \\ & & -\,{{\rm{e}}}^{(r+{r}_{0})\eta }{r}^{2}{r}_{0}{\eta }^{2}\sigma \mathrm{ExpIntegral}E{\rm{i}}[-r\eta ])-{{\rm{e}}}^{-{r}_{1}\eta }{\pi }^{2}{r}^{2}\\ & & \times \,(8{\pi }^{2}+6\pi \zeta +{\zeta }^{2})(2{{\rm{e}}}^{{r}_{1}\eta }{r}_{1}\gamma -{{\rm{e}}}^{{r}_{0}\eta }{r}_{0}(-1+{r}_{\eta })\sigma \\ & & -\,\left.\left.{{\rm{e}}}^{({r}_{0}+{r}_{1})\eta }{r}_{0}{{r}_{1}}^{2}{\eta }^{2}\sigma \mathrm{ExpIntegral}E{\rm{i}}[-{r}_{1}\eta ])\right)\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\omega }_{t}(r) & = & -\frac{1}{8{\pi }^{2}{{r}_{1}}^{2}(2\pi +\zeta )(3\pi +\zeta )(4\pi +\zeta )(r\gamma +{{\rm{e}}}^{(-r+{r}_{0})\eta }{r}_{0}\sigma )}\\ & & \times \,\left[2{{\rm{e}}}^{-r\eta }{\pi }^{2}{{r}_{1}}^{2}(6\pi +\zeta )(8{\pi }^{2}+6\pi \zeta +{\zeta }^{2})\right.\\ & & \times \,({{\rm{e}}}^{r\eta }r\gamma +{{\rm{e}}}^{{r}_{0}\eta }){r}_{0}\sigma +3(2\pi +\zeta )\\ & & \left(960{r}^{2}{{r}_{1}}^{3}(3\pi +\zeta )+{{\rm{e}}}^{-r\eta }{\pi }^{2}{{r}_{1}}^{2}(8{\pi }^{2}+6\pi \zeta +{\zeta }^{2})(2{{\rm{e}}}^{r\eta }r\gamma \right.\\ & & -\,{{\rm{e}}}^{{r}_{0}\eta }{r}_{0}(-1+r\eta )\sigma -{{\rm{e}}}^{(r+{r}_{0})\eta }{r}^{2}{r}_{0}{\eta }^{2}\sigma \mathrm{ExpIntegral}E{\rm{i}}\\ & & \times \,[-r\eta ])-{{\rm{e}}}^{-{r}_{1}\eta }{\pi }^{2}{r}^{2}(8{\pi }^{2}+6\pi \zeta +{\zeta }^{2})\\ & & \times \,(2{{\rm{e}}}^{{r}_{1}\eta }{r}_{1}\gamma -{{\rm{e}}}^{{r}_{0}\eta }{r}_{0}(-1+{r}_{\eta })\sigma -{{\rm{e}}}^{({r}_{0}+{r}_{1})\eta }\\ & & \times \,\left.\left.{r}_{0}{{r}_{1}}^{2}{\eta }^{2}\sigma \mathrm{ExpIntegral}E{\rm{i}}[-{r}_{1}\eta ])\right)\right].\end{array}\end{eqnarray}$
The expression of the anisotropy ratio at the wormhole's neck is then given by $\begin{eqnarray}\frac{{\omega }_{t}({r}_{1})}{{\omega }_{r}({r}_{1})}=-\frac{2880{{r}_{1}}^{5}(2\pi +\zeta )(3\pi +\zeta )+2{{\rm{e}}}^{-{r}_{1}\eta }{\pi }^{2}{{r}_{1}}^{2}(6\pi +\zeta )(8{\pi }^{2}+6\pi \zeta +{\zeta }^{2})({{\rm{e}}}^{{r}_{1}\eta }{r}_{1}\gamma +{{\rm{e}}}^{{r}_{0}\eta }{r}_{0}\sigma )}{2(2880{{r}_{1}}^{5}(2\pi +\zeta )(3\pi +\zeta )+2{{\rm{e}}}^{-{r}_{1}\eta }{\pi }^{2}{{r}_{1}}^{2}\zeta (8{\pi }^{2}+6\pi \zeta +{\zeta }^{2})({{\rm{e}}}^{{r}_{1}\eta }{r}_{1}\gamma +{{\rm{e}}}^{{r}_{0}\eta }{r}_{0}\sigma ))}.\end{eqnarray}$
A graphical illustration of these EoS parameters as well as the anisotropy ratio is provided in the plots of figures 4 and 5. It can be easily seen that the radial EoS parameter exhibits positive conduct while the tangential pressure shows negative behavior. Also, the anisotropy ratio indicates negative behavior approaching the value − 0.5 when r1 → ∞; the same value can be achieved when the limit r1 → ∞ is applied to the expression ( $\begin{eqnarray*}\begin{array}{rcl}V{\rm{I}}Q & = & -8\pi {\displaystyle \int }_{{r}_{1}}^{{r}_{2}}\frac{-1}{5760{r}^{3}}\left[4{\pi }^{2}(r\gamma +{{\rm{e}}}^{(-r+{r}_{0})\eta }{r}_{0}\sigma )\right.\\ & & +\,\frac{1}{{{r}_{1}}^{2}(2\pi +\zeta )(3\pi +\zeta )(4\pi +\zeta )}\left(2{{\rm{e}}}^{-r\eta }{\pi }^{2}{{r}_{1}}^{2}\zeta \right.\\ & & \times \,(8{\pi }^{2}+6\pi \zeta +{\zeta }^{2})({{\rm{e}}}^{r\eta }r\gamma +{{\rm{e}}}^{{r}_{0}\eta }{r}_{0}\sigma )+3(2\pi +\zeta )\\ & & \times \,(960{r}^{2}{{r}_{1}}^{3}(3\pi +\zeta )+{{\rm{e}}}^{-r\eta }{\pi }^{2}{{r}_{1}}^{2}\\ & & \times \,(8{\pi }^{2}+6\pi \zeta +{\zeta }^{2})(2{{\rm{e}}}^{r\eta }r\gamma -{{\rm{e}}}^{{r}_{0}\eta }{r}_{0}(-1+r\eta )\sigma \\ & & -\,{{\rm{e}}}^{(r+{r}_{0})\eta }{r}^{2}{r}_{0}{\eta }^{2}\sigma {\rm{E}}{\rm{x}}{\rm{p}}{\rm{I}}{\rm{n}}{\rm{t}}{\rm{e}}{\rm{g}}{\rm{r}}{\rm{a}}l{\rm{E}}{\rm{i}}[-r\eta ])\\ & & -\,{{\rm{e}}}^{-{r}_{1}\eta }{\pi }^{2}{r}^{2}(8{\pi }^{2}+6\pi \zeta +{\zeta }^{2})(2{{\rm{e}}}^{{r}_{1}\eta }{r}_{1}\gamma \\ & & -\,{{\rm{e}}}^{{r}_{0}\eta }{r}_{0}(-1+{r}_{1}\eta )\sigma -{{\rm{e}}}^{({r}_{0}+{r}_{1})\eta }\\ & & \times \,\left.\left.{r}_{0}{{r}_{1}}^{2}{\eta }^{2}\sigma {\rm{E}}{\rm{x}}{\rm{p}}{\rm{I}}{\rm{n}}{\rm{t}}{\rm{e}}{\rm{g}}{\rm{r}}{\rm{a}}l{\rm{E}}{\rm{i}}[-{r}_{1}\eta ]))\right)\right]{\rm{d}}r\end{array}\end{eqnarray*}$
which, on integration, yields $\begin{eqnarray*}\begin{array}{rcl}V{\rm{I}}Q & = & \frac{1}{3840}\left[\frac{8{\pi }^{2}\gamma (2\pi +\zeta )}{r(3\pi +\zeta )}-\frac{{{\rm{e}}}^{(-r+{r}_{0})\eta }{\pi }^{2}{r}_{0}(2\pi +\zeta )(-3+5r\eta )\sigma }{{r}^{2}(3\pi +\zeta )}\right.\\ & & -\,\frac{10{{\rm{e}}}^{{r}_{0}\eta }{\pi }^{3}{r}_{0}{\eta }^{2}\sigma {\rm{E}}{\rm{x}}{\rm{p}}{\rm{I}}{\rm{n}}{\rm{t}}{\rm{e}}{\rm{g}}{\rm{r}}{\rm{a}}{\rm{l}}{\rm{E}}{\rm{i}}[-r\eta ]}{3\pi +\zeta }\\ & & -\,\frac{5{{\rm{e}}}^{{r}_{0}\eta }{\pi }^{2}{r}_{0}\zeta {\eta }^{2}\sigma {\rm{E}}{\rm{x}}{\rm{p}}{\rm{I}}{\rm{n}}{\rm{t}}{\rm{e}}{\rm{g}}{\rm{r}}{\rm{a}}{\rm{l}}{\rm{E}}{\rm{i}}[-r\eta ]}{3\pi +\zeta }\\ & & -\,\frac{1}{{{r}_{1}}^{2}(12{\pi }^{2}+7\pi \zeta +{\zeta }^{2})}2{{\rm{e}}}^{-{r}_{1}\eta }(-2{{\rm{e}}}^{{r}_{1}\eta }{r}_{1}(-1440\pi {{r}_{1}}^{2}+8{\pi }^{4}\gamma \\ & & -\,480{{r}_{1}}^{2}\zeta +6{\pi }^{3}\gamma \zeta +{\pi }^{2}\gamma {\zeta }^{2}+{{\rm{e}}}^{{r}_{0}\eta }{\pi }^{2}{r}_{0}(8{\pi }^{2}+6\pi \zeta +{\zeta }^{2})\\ & & \times \,(-1+{r}_{1}\eta )\sigma +{{\rm{e}}}^{({r}_{0}+{r}_{1})\eta }{\pi }^{2}{r}_{0}{{r}_{1}}^{2}(8{\pi }^{2}+6\pi \zeta +{\zeta }^{2}){\eta }^{2}\sigma \\ & & \times \,{\rm{E}}{\rm{x}}{\rm{p}}{\rm{I}}{\rm{n}}{\rm{t}}{\rm{e}}{\rm{g}}{\rm{r}}{\rm{a}}{\rm{l}}{\rm{E}}{\rm{i}}[-{r}_{1}\eta ]{\rm{log}}[r]-\frac{1}{3\pi +\zeta }{{\rm{e}}}^{{r}_{0}\eta }{\pi }^{2}{r}_{0}(2\pi +\zeta ){\eta }^{2}\sigma \\ & & \times \,(2r\eta {\rm{H}}{\rm{y}}{\rm{p}}{\rm{e}}{\rm{r}}{\rm{g}}{\rm{e}}{\rm{o}}{\rm{m}}{\rm{e}}{\rm{t}}{\rm{r}}{\rm{i}}{\rm{c}}PFQ[1,1,1,2,2,2,-r\eta ]\\ & & +\,{\rm{log}}[r](-2{\rm{E}}{\rm{u}}{\rm{l}}{\rm{e}}{\rm{r}}{\rm{G}}{\rm{a}}{\rm{m}}{\rm{m}}{\rm{a}}-2{\rm{E}}{\rm{x}}{\rm{p}}{\rm{I}}{\rm{n}}{\rm{t}}{\rm{e}}{\rm{g}}{\rm{r}}{\rm{a}}{\rm{l}}{\rm{E}}{\rm{i}}[-r\eta ]\\ & & {\left.-\,2{\rm{G}}{\rm{a}}{\rm{m}}{\rm{m}}{\rm{a}}[0,r\eta ]+{\rm{log}}[r]-2{\rm{log}}[r\eta ]))\right]}_{{r}_{1}}^{{r}_{2}}.\end{array}\end{eqnarray*}$
Further, we analyze the stability of wormhole configuration through the TOV condition ( $\begin{eqnarray}\begin{array}{rcl}{F}_{h} & = & -\frac{{\rm{d}}{{\rm{p}}}_{r}}{{\rm{d}}r}=\frac{{{\rm{e}}}^{-(2r+{r}_{1})\eta }}{5760{r}^{6}{{r}_{1}}^{2}(3\pi +\zeta )(4\pi +\zeta )}\\ & & \times \,\left[-2{{\rm{e}}}^{(2r+{r}_{1})\eta }r{r}_{1}(12960\pi r{{r}_{1}}^{2}+24{\pi }^{4}(-3r+4{r}_{1})\gamma +4320r{{r}_{1}}^{2}\zeta \right.\\ & & +\,2{\pi }^{3}(-27r+44{r}_{1})\gamma \zeta +{\pi }^{2}(-9r+16{r}_{1})\gamma {\zeta }^{2})\\ & & -\,9{{\rm{e}}}^{(2r+{r}_{0})\eta }{\pi }^{2}{r}^{2}{r}_{0}(2\pi +\zeta )(4\pi +\zeta )(-1+{r}_{1}\eta )\sigma \\ & & +\,{{\rm{e}}}^{(r+{r}_{0}+{r}_{1})\eta }{\pi }^{2}{r}_{0}{{r}_{1}}^{2}(4\pi +\zeta )(6\pi (-5+3r\eta )+\zeta (-25+7r\eta )\sigma \\ & & +\,9{{\rm{e}}}^{(2r+{r}_{0}+{r}_{1})\eta }{\pi }^{2}{r}^{2}{r}_{0}{{r}_{1}}^{2}(2\pi +\zeta )(4\pi +\zeta )\\ & & \times \,\left.{\eta }^{2}\sigma ({\rm{E}}{\rm{x}}{\rm{p}}{\rm{I}}{\rm{n}}{\rm{t}}{\rm{e}}{\rm{g}}{\rm{r}}{\rm{a}}{\rm{l}}{\rm{E}}{\rm{i}}[-r\eta ]-{\rm{E}}{\rm{x}}{\rm{p}}{\rm{I}}{\rm{n}}{\rm{t}}{\rm{e}}{\rm{g}}{\rm{r}}{\rm{a}}{\rm{l}}{\rm{E}}{\rm{i}}[-{r}_{1}\eta ])\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{F}_{{\rm{a}}} & = & 2\frac{{{\rm{p}}}_{{\rm{t}}}-{{\rm{p}}}_{{r}}}{r}=\frac{{{\rm{e}}}^{-(r+{r}_{1})\eta }}{1920{r}^{6}{{r}_{1}}^{2}(3\pi +\zeta )(4\pi +\zeta )}\\ & & \times \,\left[2{{\rm{e}}}^{(r+{r}_{1})\eta }r{r}_{1}(4320\pi r{{r}_{1}}^{2}+8{\pi }^{4}(-3r+4{r}_{1})\gamma +1440r{{r}_{1}}^{2}\zeta \right.\\ & & +\,6{\pi }^{3}(-3r+4{r}_{1})\gamma \zeta +{\pi }^{2}(-3r+4{r}_{1})\gamma {\zeta }^{2})-{{\rm{e}}}^{({r}_{0}+{r}_{1})\eta }{\pi }^{2}{r}_{0}{{r}_{1}}^{2}(2\pi +\zeta )(4\pi +\zeta )(-5+3r\eta )\sigma \\ & & +\,3{{\rm{e}}}^{(r+{r}_{0})\eta }{\pi }^{2}{r}^{2}{{r}}_{0}(2\pi +\zeta )(4\pi +\zeta )(-1+{r}_{1}\eta )\\ & & \times \,\sigma -3{{\rm{e}}}^{(r+{r}_{0}+{r}_{1})\eta }{\pi }^{2}{r}^{2}{r}_{0}{r}_{1}^{2}(2\pi +\zeta )(4\pi +\zeta ){\eta }^{2}\sigma \\ & & \times \,\left.({\rm{E}}{\rm{x}}{\rm{p}}{\rm{I}}{\rm{n}}{\rm{t}}{\rm{e}}{\rm{g}}{\rm{r}}{\rm{a}}{\rm{l}}{\rm{E}}{\rm{i}}[-r\eta ]-{\rm{E}}{\rm{x}}{\rm{p}}{\rm{I}}{\rm{n}}{\rm{t}}{\rm{e}}{\rm{g}}{\rm{r}}{\rm{a}}{\rm{l}}{\rm{E}}{\rm{i}}[-{r}_{1}\eta ])\right].\end{array}\end{eqnarray}$
It is worth mentioning here that Fg vanishes due to the zero tidal force condition. A graphical illustration of the VIQ expression as well as the hydrostatic and anisotropic forces is provided in the plots of figure 6. From the left plot, it can be seen that the VIQ exhibits negative behavior and in the limit r2 → r1, the VIQ vanishes. Thus the VIQ ensures the involvement of exotic material, which is regarded as a compulsory element in wormhole construction. From the right panel, it is evident that the two forces exhibit equal but opposite behavior and hence balance each other's impact, thus leaving the wormhole configuration in a stable state.
Figure 4. Plots refer to the dynamics of EoS parameters in radial and tangential directions. |
Figure 5. Plots portray the conduct of the pressure anisotropy ratio $\frac{{\omega }_{t}}{{\omega }_{r}}$ and the anisotropy parameter Δ versus r. |
Figure 6. Left and right plots indicate the behavior of the VIQ expression and stability through forces of the TOV equation, respectively. |
Next, the active mass density, in this case, turns out as 21 ) results in the following expression after integration:
$\begin{eqnarray}{{\mathbb{M}}}_{A}=-\frac{{{\rm{e}}}^{-r\eta }{\pi }^{3}\left(-2{{\rm{e}}}^{r\eta }r\gamma +{{\rm{e}}}^{{r}_{0}\eta }{r}_{0}(-1+r\eta )\sigma +{{\rm{e}}}^{(r+{r}_{0})\eta }{r}^{2}{r}_{0}{\eta }^{2}\sigma {\rm{E}}{\rm{x}}{\rm{p}}{\rm{I}}{\rm{n}}{\rm{t}}{\rm{e}}{\rm{g}}r{\rm{a}}{\rm{l}}{\rm{E}}{\rm{i}}[-r\eta ]\right)}{720{r}^{2}}{| }_{{r}_{2}}^{R}.\end{eqnarray}$
Also, the complexity factor defined in equation ( $\begin{eqnarray*}\begin{array}{rcl}{Y}_{TF} & = & 8\pi ({{\rm{p}}}_{r}-{{\rm{p}}}_{{\rm{t}}})-\frac{1}{2{r}^{3}}\left[\frac{{{\rm{e}}}^{-r\eta }{\pi }^{2}\left(-8{{\rm{e}}}^{r\eta }r\gamma +{{\rm{e}}}^{{r}_{0}\eta }{r}_{0}(-5+3r\eta )\sigma +3{{\rm{e}}}^{(r+{r}_{0})\eta }{r}^{2}{r}_{0}{\eta }^{2}\sigma {\rm{E}}{\rm{x}}{\rm{p}}{\rm{I}}{\rm{n}}{\rm{t}}{\rm{e}}{\rm{g}}{\rm{r}}{\rm{a}}{\rm{l}}{\rm{E}}{\rm{i}}[-r\eta ]\right)}{2880{r}^{2}}\right.\\ & & -\,\left.\frac{{{\rm{e}}}^{-{r}_{2}\eta }{\pi }^{2}\left(-8{{\rm{e}}}^{{r}_{2}\eta }{r}_{2}\gamma +{{\rm{e}}}^{{r}_{0}\eta }{r}_{0}(-5+3{r}_{2}\eta )\sigma +3{{\rm{e}}}^{({r}_{0}+{{\rm{r}}}_{2})\eta }{r}_{0}{{r}_{2}}^{2}{\eta }^{2}\sigma {\rm{E}}{\rm{x}}{\rm{p}}{\rm{I}}{\rm{n}}{\rm{t}}{\rm{e}}{\rm{g}}{\rm{r}}{\rm{a}}{\rm{l}}{\rm{E}}{\rm{i}}[-{r}_{2}\eta ]\right)}{2880{{r}_{2}}^{2}}\right].\end{array}\end{eqnarray*}$
The plots of figure 7 provide a graphical illustration of the exoticity factor (20 ) and complexity factor for the obtained solution. It is seen that for different ζ choices, the exoticity factor indicates only negative behavior, which supports the involvement of exotic material at the wormhole's neck, while the involvement of normal matter is far from the throat. The right panel refers to the complexity factor, which shows that this factor assumes negative increasing values versus r and in the limit r → 0, one obtains YTF → 0, as expected. In the left plot of figure 8, the behavior of active gravitational mass can be observed. It is seen that for different ζ choices, ${{\mathbb{M}}}_{A}$ exhibits negative behavior, which also supports the involvement of exotic material. From its right graph, the behavior of the adiabatic index can be checked. It is easy to see that this index takes values more than $\frac{4}{3}$ for all choices − 3≤ζ≤9 and hence shows stable behavior against r. However, for some values, i.e. 9 < ζ < 10, this factor can take values less than $\frac{4}{3}$ and hence refers to instability of the wormhole geometry there.
Figure 7. Left and right plots explain the dynamics of exoticity and complexity factors, respectively. |
Figure 8. Plots on the left and right side correspond to graphical description of the active gravitational mass and adiabatic index, respectively. |
3.2. Yukawa-corrected Casimir wormhole in conformal ${\mathbb{F}}(R,{ \mathcal T })$ theory
For a given metric element, the conformal symmetries not only yield the respective constants of motion but also play a vital role in fixing the unknowns present in the dynamical equations of GR or MGFs. In this respect, the conformal symmetries can help to reduce the complexity of any system of field equations. For a specified spacetime, the CKVs can be computed through the relation:
$\begin{eqnarray*}{{ \mathcal L }}_{\psi }{g}_{\mu \nu }={g}_{\eta \nu }{\psi }_{;\mu }^{\eta }+{g}_{\mu \nu }{\psi }_{;\nu }^{\eta }=\chi (r){g}_{\mu \nu }.\end{eqnarray*}$
Herein, the unknown χ refers to the conformal vector. For a metric exhibiting spherical symmetry, the above equation results in: $\begin{eqnarray}{\psi }^{1}{\alpha }^{{\prime} }(r)=\chi (r),\quad {\psi }^{1}=\frac{r\chi (r)}{2},\quad {\psi }^{1}{\beta }^{{\prime} }+2{\psi }_{,1}^{1}=\chi (r)\end{eqnarray}$
whose integration can lead to the expressions given by ${{\rm{e}}}^{\alpha (r)}={A}^{2}{r}^{2},\,{{\rm{e}}}^{\beta (r)}={\left(\frac{B}{\chi }\right)}^{2}$. Here, A and B correspond to integration constants. Consequently, the state variables, i.e. density, radial and tangential pressures, can be isolated as follows $\begin{eqnarray}\begin{array}{rcl}\rho (r) & = & -\frac{1}{12{B}^{2}{r}^{2}(\zeta +2\pi )(\zeta +4\pi )}\\ & & \times \,\left[\right.-4{B}^{2}(\zeta +3\pi )+r(7\zeta +24\pi )\chi (r)\chi ^{\prime} (r)+3(\zeta +4\pi )\chi {(r)}^{2}\left]\right.,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{p}_{r}(r) & = & \frac{1}{12{B}^{2}{r}^{2}(\zeta +2\pi )(\zeta +4\pi )}\\ & & \times \,\left[\right.-4{B}^{2}(\zeta +3\pi )-5r\zeta \chi (r)\chi ^{\prime} (r)+3(\zeta +4\pi )\chi {(r)}^{2}\left]\right.,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{p}_{t}(r) & = & \frac{1}{12{B}^{2}{r}^{2}(\zeta +2\pi )(\zeta +4\pi )}\\ & & \times \,\left[\right.2{B}^{2}\zeta +r(7\zeta +24\pi )\chi (r)\chi ^{\prime} (r)+3(\zeta +4\pi )\chi {(r)}^{2}\left]\right..\end{array}\end{eqnarray}$
In order to get an expression for χ, we set the density given in equation (33 ) equal to the Yukawa-corrected energy density given by equation (14), as follows
$\begin{eqnarray*}\begin{array}{rcl} & & -\frac{{\pi }^{2}}{2r(720{r}^{4})}\left(\gamma r+\sigma {r}_{0}{{\rm{e}}}^{-\eta (r-{r}_{0})}\right)=-\frac{1}{12{B}^{2}{r}^{2}(\zeta +2\pi )(\zeta +4\pi )}\\ & & \times \,\left[\right.-4{B}^{2}(\zeta +3\pi )+r(7\zeta +24\pi )\chi (r)\chi ^{\prime} (r)+3(\zeta +4\pi )\chi {(r)}^{2}\left]\right..\end{array}\end{eqnarray*}$
Its simplification gives us an ODE in χ which can be solved as follows $\begin{eqnarray*}\begin{array}{rcl}\chi (r) & = & \pm \frac{1}{4{r}^{2}\sqrt{30\eta (3\pi +\zeta )(4\pi +\zeta )(24\pi +7\zeta )}}\\ & & \times \,\left[\Space{0ex}{3ex}{0ex}(-24\pi +7\zeta )\eta \left({B}^{2}{r}^{2}(32{\pi }^{5}\gamma -3840\pi {r}^{2}\zeta +32{\pi }^{4}\gamma \zeta \right.\right.\\ & & -\,640{r}^{2}{\zeta }^{2}+10{\pi }^{3}\gamma {\zeta }^{2}+{\pi }^{2}(-5760{r}^{2}+\gamma ){\zeta }^{3})\\ & & -\,\left.480{r}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}(12{\pi }^{2}+7\pi \zeta +{\zeta }^{2}){C}_{2}\right)\\ & & -\,8{B}^{2}{{\rm{e}}}^{{r}_{0}\eta }{\pi }^{2}{r}_{0}{(4\pi +\zeta )}^{2}(6{\pi }^{2}+5\pi \zeta +{\zeta }^{2})\\ & & \times \,\left.{(r\eta )}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}\sigma { \mathcal G }\left[-\frac{3(16\pi +5\zeta )}{24\pi +7\zeta },r\eta \right]\right],\end{array}\end{eqnarray*}$
where ${ \mathcal G }$ represents the Gamma function and ${{ \mathcal C }}_{2}$ is an integration constant. Using this value, the shape model can be found as $\begin{eqnarray*}\begin{array}{rcl}S(r) & = & \frac{1}{480{B}^{2}{r}^{3}(288{\pi }^{3}+252{\pi }^{2}\zeta +73\pi {\zeta }^{2}+7{\zeta }^{3})\eta }\left[\Space{0ex}{3ex}{0ex},[,\right.(24\pi +7\zeta )\eta \left({B}^{2}{r}^{2}(32{\pi }^{5}\gamma -480\pi {r}^{2}\zeta +32{\pi }^{4}\gamma \zeta \right.\\ & & -\,160{r}^{2}{\zeta }^{2}+10{\pi }^{3}\gamma {\zeta }^{2}+{\pi }^{2}\gamma {\zeta }^{3})-480{r}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}(12{\pi }^{2}+7\pi \zeta +{\zeta }^{2}){C}_{2})+8{B}^{2}{{\rm{e}}}^{{r}_{0}\eta }{\pi }^{2}{r}_{0}{(4\pi +\zeta )}^{2}\\ & & \times \,\left.\left(6{\pi }^{2}+5\pi \zeta +{\zeta }^{2}){(r\eta )}^{\frac{72\pi +2\zeta }{24\pi +7\zeta }}\sigma { \mathcal G }[-\frac{3(16\pi +5\zeta )}{24\pi +7\zeta },r\eta \right]\right).\end{array}\end{eqnarray*}$
In order to determine this integration constant, we utilize the condition S(r1) = r1. Following similar lines of the previous segment, the integration constant can be found as follows $\begin{eqnarray*}\begin{array}{rcl}{{ \mathcal C }}_{2} & = & \frac{1}{480(24\pi +7\zeta )(12{\pi }^{2}+7\pi \zeta +{\zeta }^{2})\eta }\left[\Space{0ex}{2.5ex}{0ex}{B}^{2}{{r}_{1}}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}\left({{r}_{1}}^{2}(24\pi +7\zeta )(32{\pi }^{5}\gamma -3840\pi {{r}_{1}}^{2}\zeta +32{\pi }^{4}\gamma \zeta \right.\right.\\ & & -\,640{{r}_{1}}^{2}{\zeta }^{2}+10{\pi }^{3}\gamma {\zeta }^{2}+{\pi }^{2}(-5760{{r}_{1}}^{2}+\gamma {\zeta }^{3}))\eta +8{{\rm{e}}}^{{r}_{0}\eta }{\pi }^{2}{r}_{0}{(4\pi +\zeta )}^{2}(6{\pi }^{2}+5\pi \zeta +{\zeta }^{2}){({r}_{1}\eta )}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}\\ & & \times \,\left.\left.\sigma { \mathcal G }[-\frac{3(16\pi +5\zeta )}{24\pi +7\zeta },{r}_{1}\eta ]\right)\right].\end{array}\end{eqnarray*}$
Consequently, the final form of the shape model can be obtained as $\begin{eqnarray*}\begin{array}{rcl}S(r) & = & \frac{1}{480{B}^{2}{r}^{3}(288{\pi }^{3}+252{\pi }^{2}\zeta +73\pi {\zeta }^{2}+7{\zeta }^{3})\eta }\left[\Space{0ex}{3.5ex}{0ex}{B}^{2}{r}^{2}(24\pi +7\zeta )(32{\pi }^{5}\gamma -480\pi {r}^{2}\zeta +32{\pi }^{4}\gamma \zeta \right.\\ & & -\,160{r}^{2}{\zeta }^{2}+10{\pi }^{3}\gamma {\zeta }^{2}+{\pi }^{2}\gamma {\zeta }^{3})\eta +8{B}^{2}{{\rm{e}}}^{{r}_{0}\eta }{\pi }^{2}{r}_{0}{(4\pi +\zeta )}^{2}(6{\pi }^{2}+5\pi \zeta +{\zeta }^{2}){(r\eta )}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}\\ & & \times \,\sigma { \mathcal G }[-\frac{3(16\pi +5\zeta )}{24\pi +7\zeta },r\eta ]-{B}^{2}{r}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{-72\pi +22\zeta 24\pi +7\zeta }\left({{r}_{1}}^{2}(24\pi +7\zeta )(32{\pi }^{5}\gamma -3840\pi {{r}_{1}}^{2}\zeta \right.\\ & & +\,32{\pi }^{4}\gamma \zeta -640{{r}_{1}}^{2}{\zeta }^{2}+10{\pi }^{3}\gamma {\zeta }^{2}+{\pi }^{2}(-5760{{r}_{1}}^{2}+\gamma {\zeta }^{3}))\eta +8{{\rm{e}}}^{{r}_{0}\eta }{\pi }^{2}{r}_{0}{(4\pi +\zeta )}^{2}(6{\pi }^{2}+5\pi \zeta +{\zeta }^{2})\\ & & \times \,\left.\left.{({r}_{1}\eta )}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}\sigma { \mathcal G }[-\frac{3(16\pi +5\zeta )}{24\pi +7\zeta },{r}_{1}\eta ]\right)\right].\end{array}\end{eqnarray*}$
The graphical analysis of essential properties of this wormhole shape model is presented in the plots of figures 9 and 10. It can be seen that S(r) exhibits positive but decreasing behavior for positive ζ choices, while increasing behavior can be achieved for negative ζ values. The function $S^{\prime} (r)$ shows negative behavior for ζ > 0 while a positive behavior can be observed for ζ < 0; in both cases, however, the condition $S^{\prime} ({r}_{1})\lt 1$ remains consistent. Next, the left plot of figure 10 indicates that the obtained wormhole shape model does not meet the asymptotic flatness requirement as $1-\frac{S}{r}\approx 0.9$ as r → 10. However, for values r > 10, this condition may be fulfilled. The right plot shows that constraint S(r) − r < 0 is compatible in the present case.
Figure 9. Plots (a) and (b), respectively, provide the dynamics of S(r) and its derivative against r for a Yukawa-corrected wormhole with conformal symmetries. Herein, we fix: r0 = 2, γ = 2.5, σ = − 2, η = 1, r1 = 2 and B = 1. |
Figure 10. Left and right plots, respectively, refer to the graphical illustration of of conditions $1-\frac{S(r)}{r}$ and S(r) − r. |
The values of the radial and tangential components of pressure are given by
$\begin{eqnarray*}\begin{array}{rcl}{p}_{r} & = & \frac{{{\rm{e}}}^{-r\eta }{{r}_{1}}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}}{1440{r}^{6}(3\pi +\zeta )(4\pi +\zeta ){(24\pi +7\zeta )}^{2}\eta }\left[\Space{0ex}{2.5ex}{0ex}(24\pi +7\zeta )\eta ({{\rm{e}}}^{r\eta }\left(-288{\pi }^{5}(-{r}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{2}+{r}^{2}{{r}_{1}}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }})\gamma \right.\right.\\ & & -\,34560\pi {r}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{4}\zeta -12{\pi }^{4}(-24{r}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{2}+29{r}^{2}{{r}_{1}}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}\gamma \zeta )-5760{r}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{4}{\zeta }^{2}\\ & & -\,\left.5{\pi }^{3}(-18{r}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{2}+25{r}^{2}{{r}_{1}}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}\gamma {\zeta }^{2})+{\pi }^{2}(-14{r}^{2}{{r}_{1}}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}\gamma {\zeta }^{3}+{r}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}(-51840{{r}_{1}}^{4}+9{{r}_{1}}^{2}\gamma {\zeta }^{3}))\right)\\ & & -\,5{{\rm{e}}}^{{r}_{0}\eta }{\pi }^{2}r{r}_{0}{{r}_{1}}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}\zeta (12{\pi }^{2}+7\pi \zeta +{\zeta }^{2})\sigma )-72{{\rm{e}}}^{(r+{r}_{0})\eta }{\pi }^{2}{r}_{0}{{r}_{1}}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}{(4\pi +\zeta )}^{2}(6{\pi }^{2}+5\pi \zeta +{\zeta }^{2})\\ & & \times \,{(r\eta )}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}\sigma { \mathcal G }[-\frac{3(16\pi +5\zeta )}{24\pi +7\zeta },r\eta ]+72{{\rm{e}}}^{(r+{r}_{0})\eta }{\pi }^{2}{r}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}{r}_{0}{(4\pi +\zeta )}^{2}(6{\pi }^{2}+5\pi \zeta +{\zeta }^{2})\\ & & \times \,\left.{({r}_{1}\eta )}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}\sigma { \mathcal G }[-\frac{3(16\pi +5\zeta )}{24\pi +7\zeta },{r}_{1}\eta ]\right],\\ {p}_{t} & = & \frac{\frac{720{r}^{3}}{4\pi +\zeta }+{\pi }^{2}\left(r\gamma +{{\rm{e}}}^{(-r+{r}_{0})\eta }{r}_{0}\sigma \right)}{1440{r}^{5}}.\end{array}\end{eqnarray*}$
Further, the EoS parameters for the radial and tangential pressures can be obtained as $\begin{eqnarray*}\begin{array}{rcl}{\omega }_{r} & = & -\frac{2{{\rm{e}}}^{-r\eta }{{r}_{1}}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}}{{\pi }^{2}r(3\pi +\zeta )(4\pi +\zeta ){(24\pi +7\zeta )}^{2}\eta (r\gamma +{{\rm{e}}}^{(-r+{r}_{0})\eta }{r}_{0}\sigma )}\left[\Space{0ex}{1ex}{0ex}(24\pi +7\zeta )\eta \left({{\rm{e}}}^{r\eta }(-288{\pi }^{5}(-{r}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{2}\right.\right.\\ & & +\,{r}^{2}{{r}_{1}}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }})\gamma -34560\pi {r}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{4}\zeta -12{\pi }^{4}(-24{r}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{2}+29{r}^{2}{{r}_{1}}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }})\gamma \zeta -5760{r}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{4}{\zeta }^{2}\\ & & -\,5{\pi }^{3}(-18{r}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{2}+25{r}^{2}{{r}_{1}}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }})\gamma {\zeta }^{2}+{\pi }^{2}(-14{r}^{2}{{r}_{1}}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}\gamma {\zeta }^{3}+{r}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}(-51840{{r}_{1}}^{4}+9{{r}_{1}}^{2}\gamma {\zeta }^{3})))\\ & & -\,\left.5{{\rm{e}}}^{{r}_{0}\eta }{\pi }^{2}r{r}_{0}{{r}_{1}}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}\zeta (12{\pi }^{2}+7\pi \zeta +{\zeta }^{2})\sigma \right)-72{{\rm{e}}}^{(r+{r}_{0})\eta }{\pi }^{2}{r}_{0}{{r}_{1}}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}{(4\pi +\zeta )}^{2}(6{\pi }^{2}+5\pi \zeta +{\zeta }^{2}){(r\eta )}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}\\ & & \times \,\sigma { \mathcal G }[-\frac{3(16\pi +5\zeta )}{(24\pi +7\zeta },r\eta ]+72{{\rm{e}}}^{(r+{r}_{0})\eta }{\pi }^{2}{r}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}{r}_{0}{(4\pi +\zeta )}^{2}(6{\pi }^{2}+5\pi \zeta +{\zeta }^{2}){({r}_{1}\eta )}^{\frac{72\pi +22\zeta }{24\pi +7\zeta }}\\ & & \times \,\sigma { \mathcal G }[-\frac{3(16\pi +5\zeta )}{(24\pi +7\zeta },{r}_{1}\eta ]\Space{0ex}{3ex}{0ex}],\\ {\omega }_{t} & = & -2-\frac{1440{r}^{3}}{{\pi }^{2}(4\pi +\zeta )(r\gamma +{{\rm{e}}}^{(-r+{r}_{0})\eta }{r}_{0}\sigma )}.\end{array}\end{eqnarray*}$
The value of pressure anisotropy at the wormhole's neck can be written as $\begin{eqnarray*}\frac{{\omega }_{t}({r}_{1})}{{\omega }_{r}({r}_{1})}=-\frac{{{\rm{e}}}^{{r}_{1}\eta }(24\pi +7\zeta )\left(720{{r}_{1}}^{3}+{\pi }^{2}{r}_{1}\gamma (4\pi +\zeta )+{{\rm{e}}}^{({r}_{0}-{r}_{1})\eta }{\pi }^{2}{r}_{0}(4\pi +\zeta )\sigma \right)}{5\left({{\rm{e}}}^{{r}_{1}\eta }{r}_{1}(3456\pi {{r}_{1}}^{2}+1152{{r}_{1}}^{2}\zeta +4{\pi }^{3}\gamma \zeta +{\pi }^{2}\gamma {\zeta }^{2})+{{\rm{e}}}^{{r}_{0}\eta }{\pi }^{2}{r}_{0}\zeta (4\pi +\zeta )\sigma \right)}.\end{eqnarray*}$
The validity of the NEC inequalities is explored in the plots of figure 11. It can be easily checked that the NEC for radial pressure is not valid near as well as far from the throat for 10≤ζ < − 9; meanwhile, the NEC corresponding to tangential pressure holds for positive ζ values, while for negative values of ζ, it is invalid. The graphs of EoS parameters along with the pressure anisotropy can be seen in figures 12 and 13. It is evident that EoS for radial pressure exhibits positive behavior while the EoS parameter for tangential pressure shows negative behavior against r. Also, the pressures anisotropy function exhibits linear, negative and decreasing behavior. It can be checked that in the limit r1 → ∞, one may obtain $-\frac{24\pi +7\zeta }{8(3\pi +\zeta )}$ and in the case ζ = 1, it yields a value of − 0.99, hence confirming the involvement of exotic matter. In the right plot of figure 13, the conduct of the anisotropy parameter is provided which shows the negative behavior for all ζ values and hence leads to the attractive nature of gravity.
Figure 11. Graphs portraying the dynamics of inequalities of NEC for both radial and tangential pressures. |
Figure 12. Graphs indicating the behavior of EoS parameters for radial and tangential pressures. |
Figure 13. Graphs revealing the graphical conduct of the anisotropy ratio $\frac{{\omega }_{t}}{{\omega }_{r}}$ and Δ against r. |
Further, the VIQ for the proposed solution can be computed as follows
$\begin{eqnarray*}\begin{array}{rcl}V{\rm{I}}Q & = & -\frac{{{\rm{e}}}^{-r+{r}_{0}}{\pi }^{2}{r}_{0}(2\pi +\zeta )(4\pi +\zeta )(-6+2r-{r}^{2}+{r}^{3}+{{\rm{e}}}^{r}{r}^{4}{\rm{E}}{\rm{x}}{\rm{p}}{\rm{I}}{\rm{n}}{\rm{t}}e{\rm{g}}r{\rm{a}}{\rm{l}}{\rm{E}}{\rm{i}}[-r])}{240{r}^{4}(24\pi +7\zeta )(4\pi +13\zeta ))}\\ & & -\,\frac{1}{34560(3\pi +\zeta )(4\pi +\zeta )}\left[-\frac{12{\pi }^{2}\gamma (12{\pi }^{2}+7\pi \zeta +{\zeta }^{2})}{r}\right.\\ & & -\,\frac{2{{\rm{e}}}^{-r+{r}_{0}}{\pi }^{2}{r}_{0}(144\pi (-1+r){r}^{2}+(-30+10r-47{r}^{2}+47{r}^{3})\zeta )(12{\pi }^{2}+7\pi \zeta +{\zeta }^{2})}{{r}^{4}(24\pi +7\zeta )}\\ & & -\,\frac{8{\pi }^{2}\gamma (12{\pi }^{2}+11\pi \zeta +2{\zeta }^{2})}{{r}^{3}}-\frac{82944{{\rm{e}}}^{{r}_{0}}{\pi }^{6}{r}_{0}{\rm{E}}{\rm{x}}{\rm{p}}{\rm{I}}{\rm{n}}{\rm{t}}e{\rm{g}}r{\rm{a}}{\rm{l}}{\rm{E}}{\rm{i}}[-r]}{{(24\pi +7\zeta )}^{2}}-\frac{99648{{\rm{e}}}^{{r}_{0}}{\pi }^{5}{r}_{0}\zeta {\rm{E}}{\rm{x}}{\rm{p}}{\rm{I}}{\rm{n}}{\rm{t}}e{\rm{g}}r{\rm{a}}{\rm{l}}{\rm{E}}{\rm{i}}[-r]}{{(24\pi +7\zeta )}^{2}}\\ & & -\,\frac{44712{{\rm{e}}}^{{r}_{0}}{\pi }^{4}{r}_{0}{\zeta }^{2}{\rm{E}}{\rm{x}}{\rm{p}}{\rm{I}}{\rm{n}}{\rm{t}}e{\rm{g}}r{\rm{a}}{\rm{l}}{\rm{E}}{\rm{i}}[-r]}{{(24\pi +7\zeta )}^{2}}-\frac{8878{{\rm{e}}}^{{r}_{0}}{\pi }^{3}{r}_{0}{\zeta }^{3}{\rm{E}}{\rm{x}}{\rm{p}}{\rm{I}}{\rm{n}}{\rm{t}}e{\rm{g}}r{\rm{a}}{\rm{l}}{\rm{E}}{\rm{i}}[-r]}{{(24\pi +7\zeta )}^{2}}-\frac{658{{\rm{e}}}^{{r}_{0}}{\pi }^{2}{r}_{0}{\zeta }^{4}{\rm{E}}{\rm{x}}{\rm{p}}{\rm{I}}{\rm{n}}{\rm{t}}e{\rm{g}}r{\rm{a}}{\rm{l}}{\rm{E}}{\rm{i}}[-r]}{{(24\pi +7\zeta )}^{2}}\\ & & +\,\frac{3456{{\rm{e}}}^{{r}_{0}}{\pi }^{2}{r}^{-\frac{48\pi +13\zeta }{24\pi +7\zeta }}{r}_{0}{(4\pi +\zeta )}^{2}(6{\pi }^{2}+5\pi \zeta +{\zeta }^{2}){ \mathcal G }[-\frac{3(16\pi +5\zeta )}{24\pi +7\zeta },r]}{(24\pi +7\zeta )(48\pi +13\zeta )}+\frac{216{r}^{-\frac{48\pi +13\zeta }{24\pi +7\zeta }}{{r}_{1}}^{\frac{-72\pi +22\zeta }{24\pi +7\zeta }}}{(24\pi +7\zeta )(48\pi +13\zeta )}\\ & & \times \,\left({{r}_{1}}^{2}(24\pi +7\zeta )(32{\pi }^{5}\gamma -3840\pi {{r}_{1}}^{2}\zeta +32{\pi }^{4}\gamma \zeta -640{{r}_{1}}^{2}{\zeta }^{2}+10{\pi }^{3}\gamma {\zeta }^{2}+{\pi }^{2}(-5760{{r}_{1}}^{2}+\gamma {\zeta }^{3}))\right.\\ & & -\,{\left.\left.16{{\rm{e}}}^{{r}_{0}}{\pi }^{2}{r}_{0}{{r}_{1}}^{-\frac{48\pi +13\zeta }{24\pi +7\zeta }}{(4\pi +\zeta )}^{2}(6{\pi }^{2}+5\pi \zeta +{\zeta }^{2}){ \mathcal G }[-\frac{3(16\pi +5\zeta )}{24\pi +7\zeta },{r}_{1}]\right)\right]}_{{r}_{0}}^{{r}_{1}}.\end{array}\end{eqnarray*}$
In the present case, the complexity factor turns out as
$\begin{eqnarray*}\begin{array}{rcl}{Y}_{TF} & = & 8\pi ({{\rm{p}}}_{r}-{{\rm{p}}}_{{\rm{t}}})-\frac{1}{2{{\rm{r}}}^{3}}\left[\right.\frac{{{\rm{e}}}^{-{\rm{r}}\eta }{\pi }^{2}\left(-8{{\rm{e}}}^{{\rm{r}}\eta }{\rm{r}}\gamma +{{\rm{e}}}^{{{\rm{r}}}_{0}\eta }{{\rm{r}}}_{0}(-5+3{\rm{r}}\eta )\sigma +3{{\rm{e}}}^{({\rm{r}}+{{\rm{r}}}_{0})\eta }{{\rm{r}}}^{2}{{\rm{r}}}_{0}{\eta }^{2}\sigma {\rm{E}}{\rm{x}}{\rm{p}}{\rm{I}}{\rm{n}}{\rm{t}}eg{\rm{r}}{\rm{a}}{\rm{l}}{\rm{E}}{\rm{i}}[-{\rm{r}}\eta ]\right)}{2880{{\rm{r}}}^{2}}\\ & & -\,\frac{{{\rm{e}}}^{-{{\rm{r}}}_{2}\eta }{\pi }^{2}\left(-8{{\rm{e}}}^{{{\rm{r}}}_{2}\eta }{{\rm{r}}}_{2}\gamma +{{\rm{e}}}^{{{\rm{r}}}_{0}\eta }{{\rm{r}}}_{0}(-5+3{{\rm{r}}}_{2}\eta )\sigma +3{{\rm{e}}}^{({{\rm{r}}}_{0}+{{\rm{r}}}_{2})\eta }{{\rm{r}}}_{0}{{{\rm{r}}}_{2}}^{2}{\eta }^{2}\sigma {\rm{E}}{\rm{x}}{\rm{p}}{\rm{I}}{\rm{n}}{\rm{t}}eg{\rm{r}}{\rm{a}}{\rm{l}}{\rm{E}}{\rm{i}}[-{{\rm{r}}}_{2}\eta ]\right)}{2880{{{\rm{r}}}_{2}}^{2}}.\end{array}\end{eqnarray*}$
Also, the forces Fg, Fh and Fa of the TOV equation take the following form:
$\begin{eqnarray*}\begin{array}{rcl}{F}_{g} & = & \frac{1}{2880{r}^{7}(3\pi +\zeta )(4\pi +\zeta ){(24\pi +7\zeta )}^{2}}\left[\Space{0ex}{2.5ex}{0ex}-{\pi }^{2}r(2{{\rm{e}}}^{-r+{r}_{0}}{r}_{0}-r\gamma )(3\pi +\zeta )(4\pi +\zeta ){(24\pi +7\zeta )}^{2}\right.\\ & & +\,2{{\rm{e}}}^{-r}{{r}_{1}}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}(24\pi +7\zeta )(-10{{\rm{e}}}^{{r}_{0}}{\pi }^{2}r{r}_{0}{{r}_{1}}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}\zeta (12{\pi }^{2}+7\pi \zeta +{\zeta }^{2})+{{\rm{e}}}^{r}(288{\pi }^{5}(-{r}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{2}\\ & & +\,{r}^{2}{r}_{1}-\frac{72\pi +22\zeta }{24\pi +7\zeta })\gamma +34560\pi {r}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{4}\zeta +12{\pi }^{4}(-24{r}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{2}+29{r}^{2}{{r}_{1}}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }})\gamma \zeta +5760{r}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{4}{\zeta }^{2}\\ & & +\,5{\pi }^{3}(-18{r}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{2}+25{r}^{2}{{r}_{1}}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }})\gamma {\zeta }^{2}+{\pi }^{2}(14{r}^{2}{{r}_{1}}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}\gamma {\zeta }^{3}+9{r}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{2}(5760{{r}_{1}}^{2}-\gamma {\zeta }^{3}))))\\ & & -288{{\rm{e}}}^{{r}_{0}}{\pi }^{2}{r}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}{r}_{0}{(4\pi +\zeta )}^{2}(6{\pi }^{2}+5\pi \zeta +{\zeta }^{2}){ \mathcal G }[-\frac{3(16\pi +5\zeta )}{24\pi +7\zeta },r]+288{{\rm{e}}}^{{r}_{0}}{\pi }^{2}{r}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}{r}_{0}{(4\pi +\zeta )}^{2}\\ & & \times \,\left.(6{\pi }^{2}+5\pi \zeta +{\zeta }^{2}){ \mathcal G }[-\frac{3(16\pi +5\zeta )}{24\pi +7\zeta },{r}_{1}]\right],\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{F}_{h} & = & \frac{1}{1440{r}^{7}(3\pi +\zeta )(4\pi +\zeta ){(24\pi +7\zeta )}^{3}}\left[\Space{0ex}{2.5ex}{0ex}{{\rm{e}}}^{-2r}{{r}_{1}}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}(24\pi +7\zeta )\left(2{{\rm{e}}}^{r+{r}_{0}}{\pi }^{2}r{r}_{0}{{r}_{1}}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}(12{\pi }^{2}\right.\right.\\ & & +\,7\pi \zeta +{\zeta }^{2})(576{\pi }^{2}+24\pi (43+5r)\zeta +(247+35r){\zeta }^{2})+4{{\rm{e}}}^{2r}(-1728{\pi }^{6}(-3{r}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{2}+4{r}^{2}{{r}_{1}}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}\gamma )\\ & & -\,288{\pi }^{5}(-23{r}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{2}+36{r}^{2}{{r}_{1}}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }})\gamma \zeta -276480\pi {r}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{4}{\zeta }^{2}-36{\pi }^{4}(-85{r}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{2}\\ & & +\,151{r}^{2}{{r}_{1}}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }})\gamma {\zeta }^{2}-28800{r}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{4}{\zeta }^{3}+{\pi }^{2}\zeta (-98{r}^{2}{{r}_{1}}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}\gamma {\zeta }^{3}+{r}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}(-881280{{r}_{1}}^{4}+45{{r}_{1}}^{2}\gamma {\zeta }^{3}))\\ & & +\,\left.{\pi }^{3}(-1211{r}^{2}{r}_{1}-\frac{72\pi +22\zeta }{24\pi +7\zeta }\gamma {\zeta }^{3}+{r}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}(-933120{{r}_{1}}^{4}+612{{r}_{1}}^{2}\gamma {\zeta }^{3})))\right)+576{{\rm{e}}}^{{r}_{0}}{\pi }^{2}{r}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}{r}_{0}(4\pi +{\zeta }^{2}\\ & & \times \,(108{\pi }^{3}+120{\pi }^{2}\zeta +43\pi {\zeta }^{2}+5{\zeta }^{3}){ \mathcal G }[-\frac{3(16\pi +5\zeta )}{24\pi +7\zeta },r]-576{{\rm{e}}}^{{r}_{0}}{\pi }^{2}{r}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}{r}_{0}{(4\pi +\zeta )}^{2}(108{\pi }^{3}\\ & & +\,\left.120{\pi }^{2}\zeta +43\pi {\zeta }^{2}+5{\zeta }^{3}){ \mathcal G }[-\frac{3(16\pi +5\zeta )}{24\pi +7\zeta },{r}_{1}]\right],\\ {F}_{a} & = & \frac{1}{720{r}^{7}(3\pi +\zeta )(4\pi +\zeta ){(24\pi +7\zeta )}^{2}}\left[\Space{0ex}{2.5ex}{0ex}720{r}^{4}(3\pi +\zeta ){(24\pi +7\zeta )}^{2}-{\pi }^{2}r(2{{\rm{e}}}^{-r+{r}_{0}}{r}_{0}-r\gamma )(3\pi +\zeta )\right.\\ & & \times \,(4\pi +\zeta ){(24\pi +7\zeta )}^{2}+{{\rm{e}}}^{-r}{{r}_{1}}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}(24\pi +7\zeta )\left(-10{{\rm{e}}}^{{r}_{0}}{\pi }^{2}r{r}_{0}{{r}_{1}}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}\zeta (12{\pi }^{2}+7\pi \zeta +{\zeta }^{2})\right.\\ & & +\,{{\rm{e}}}^{r}(288{\pi }^{5}(-{r}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{2}+{r}^{2}{{r}_{1}}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }})\gamma +34560\pi {r}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{4}\zeta +12{\pi }^{4}(-24{r}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{2}\\ & & +\,29{r}^{2}{{r}_{1}}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }})\gamma \zeta +5760{r}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{4}{\zeta }^{2}+5{\pi }^{3}(-18{r}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{2}+25{r}^{2}{{r}_{1}}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }})\gamma {\zeta }^{2}\\ & & +\,\left.{\pi }^{2}(14{r}^{2}{{r}_{1}}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}\gamma {\zeta }^{3}+9{r}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}{{r}_{1}}^{2}(5760{{r}_{1}}^{2}-\gamma {\zeta }^{3}))\right)-144{{\rm{e}}}^{{r}_{0}}{\pi }^{2}{r}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}{r}_{0}{(4\pi +\zeta )}^{2}\\ & & \times \,(6{\pi }^{2}+5\pi \zeta +{\zeta }^{2}){ \mathcal G }[-\frac{3(16\pi +5\zeta )}{24\pi +7\zeta },r]+144{{\rm{e}}}^{{r}_{0}}{\pi }^{2}{r}^{-\frac{72\pi +22\zeta }{24\pi +7\zeta }}{r}_{0}{(4\pi +\zeta )}^{2}(6{\pi }^{2}+5\pi \zeta +{\zeta }^{2})\\ & & \times \,\left.{ \mathcal G }[-\frac{3(16\pi +5\zeta )}{24\pi +7\zeta },{r}_{1}]\right].\end{array}\end{eqnarray*}$
The graphical illustration of the VIQ for the proposed wormhole shape model is provided in the left plot of figure 14 which shows that the VIQ takes negative values (ensuring the exotic material presence) and vanishes in the limit r2 → r1. Its right plot demonstrates the stability of the presented configuration through three forces of the TOV equation. It can be observed that all three forces have an almost equal but opposite impact versus the increasing r and hence leave the configuration in a stable form. Next, we explore the dynamics of the exoticity factor for the obtained solution which is provided in the left plot of figure 15. It is seen that this factor can achieve non-negative values only for a small range − 10≤ζ≤ − 9, while it takes negative values for all other ζ choices. It can be concluded that the exotic material is present at the wormhole's neck, but that far away from the neck, the ordinary matter is present. In the right graph, the behavior of the complexity factor is investigated, which shows a negative conduct that approaches zero when r → ∞. The adiabatic index is illustrated graphically in the figure 16 where it can be seen that this index takes values more than $\frac{4}{3}$ for all choices of ζ and hence affirms the stability of the wormhole configuration.
Figure 14. Plots on the left and right, respectively, refer to the behavior of the VIQ and the three forces of the TOV equation against r. |
Figure 15. Graphs on the left and right, respectively, demonstrate the dynamics of the exoticity and complexity factors. |
Figure 16. Graph demonstrates the dynamics of the adiabatic index versus r. |
3.3. New wormhole shape function in conformal ${\mathbb{F}}(R,{ \mathcal T })$ gravity
In this section, we shall consider a newly-proposed wormhole shape model and investigate the dynamics of the resulting wormhole geometry through graphs of different important measures in ${\mathbb{F}}(R,{ \mathcal T })$ theory with conformal symmetries. In order to get new insights of wormhole solutions in modified theories, Kiroriwal et al [97] proposed an interesting shape model by considering some previously presented wormhole functions [98, 99]; this is defined as 15 ) take the following forms:
$\begin{eqnarray}S(r)={{\rm{e}}}^{1-\frac{r}{{r}_{0}}}\cosh (\frac{r}{{r}_{0}}),\end{eqnarray}$
where r0 denotes the throat of the wormhole. From the relation ${\left(\frac{B}{\chi (r)}\right)}^{2}=\frac{1}{1-\frac{S}{r}}$, the expression of χ can be evaluated as $\begin{eqnarray*}\chi (r)=\pm \frac{\sqrt{{B}^{2}r-{B}^{2}{{\rm{e}}}^{1-\frac{r}{{r}_{0}}}\cosh \frac{r}{{r}_{0}}}}{\sqrt{r}}.\end{eqnarray*}$
Using this value, the resulting expressions for density, radial and tangential pressures are given by $\begin{eqnarray*}\begin{array}{rcl}\rho & = & -\frac{{{\rm{e}}}^{-\frac{2r}{{r}_{0}}}\left(48{\rm{e}}\pi r+{{\rm{e}}}^{1+\frac{2r}{{r}_{0}}}{r}_{0}\zeta -4{{\rm{e}}}^{\frac{2r}{{r}_{0}}}r{r}_{0}\zeta +{\rm{e}}(14r+{r}_{0})\zeta \right)}{48{r}^{3}{r}_{0}(2\pi +\zeta )(4\pi +\zeta )},\\ {p}_{r} & = & \frac{{{\rm{e}}}^{-\frac{r}{{r}_{0}}}\left(-{\rm{e}}(24\pi {r}_{0}+5r\zeta +11{r}_{0}\zeta )\cosh (\frac{r}{{r}_{0}})+r\zeta (-2{{\rm{e}}}^{\frac{r}{{r}_{0}}}{r}_{0}+5{\rm{e}}\sinh (\frac{r}{{r}_{0}}))\right)}{24{r}^{3}{r}_{0}(2\pi +\zeta )(4\pi +\zeta )},\\ {p}_{t} & = & \frac{{{\rm{e}}}^{\frac{-2r}{{r}_{0}}}\left(48{\rm{e}}\pi r+{{\rm{e}}}^{1+\frac{2r}{{r}_{0}}}{r}_{0}\zeta +{\rm{e}}(14r+{r}_{0})\zeta +4{{\rm{e}}}^{\frac{2r}{{r}_{0}}}r{r}_{0}(12\pi +5\zeta )\right)}{48{r}^{3}{r}_{0}(2\pi +\zeta )(4\pi +\zeta )}.\end{array}\end{eqnarray*}$
Also, the EoS parameters along radial and tangential directions are defined by $\begin{eqnarray*}{\omega }_{r}(r)=\frac{4{{\rm{e}}}^{\frac{2r}{{r}_{0}}}r{r}_{0}\zeta +{{\rm{e}}}^{1+\frac{2r}{{r}_{0}}}{r}_{0}(24\pi +11\zeta )+{\rm{e}}(24\pi {r}_{0}+10r\zeta +11{r}_{0}\zeta )}{48{\rm{e}}\pi r+{{\rm{e}}}^{1+\frac{2r}{{r}_{0}}}{r}_{0}\zeta -4{{\rm{e}}}^{\frac{2r}{{r}_{0}}}r{r}_{0}\zeta +{\rm{e}}(14r+{r}_{0})\zeta },\end{eqnarray*}$
$\begin{eqnarray*}{\omega }_{t}(r)=-\frac{48{\rm{e}}\pi r+{{\rm{e}}}^{1+\frac{2r}{{r}_{0}}}{r}_{0}\zeta +{\rm{e}}(14r+{r}_{0})\zeta +4{{\rm{e}}}^{\frac{2r}{{r}_{0}}}r{r}_{0}(12\pi +5\zeta )}{48{\rm{e}}\pi r+{{\rm{e}}}^{1+\frac{2r}{{r}_{0}}}{r}_{0}\zeta -4{{\rm{e}}}^{\frac{2r}{{r}_{0}}}r{r}_{0}\zeta +{\rm{e}}(14r+{r}_{0})\zeta }.\end{eqnarray*}$
The inequalities of energy conditions ( $\begin{eqnarray*}\begin{array}{rcl}\rho +{p}_{r} & = & -\frac{{{\rm{e}}}^{1-\frac{2r}{{r}_{0}}}(2r+{r}_{0}+{{\rm{e}}}^{\frac{2r}{{r}_{0}}}{r}_{0})}{4{r}^{3}{r}_{0}(4\pi +\zeta )},\quad \rho +{p}_{t}=\frac{1}{8\pi {r}^{2}+2{r}^{2}\zeta },\\ \rho +{p}_{r}+2{p}_{t} & = & \frac{{{\rm{e}}}^{-\frac{2r}{{r}_{0}}}\left(-{{\rm{e}}}^{1+\frac{2r}{{r}_{0}}}{r}_{0}(12\pi +5\zeta )+4{{\rm{e}}}^{\frac{2r}{{r}_{0}}}r{r}_{0}(12\pi +5\zeta )+{\rm{e}}(24\pi r-12\pi {r}_{0}+2r\zeta -5{r}_{0}\zeta )\right)}{24{r}^{3}{r}_{0}(2\pi +\zeta )(4\pi +\zeta )}.\end{array}\end{eqnarray*}$
The behavior of the shape model (36 ) is presented in the plots of figure 17. It can be easily noticed that all necessary properties for a valid wormhole configuration are satisfied. For example, S(r) exhibits positive behavior and its derivative is in accordance with the condition $S^{\prime} ({r}_{0})\lt 1$ and S(r) − r < 0; however, the asymptotic flat condition is not achieved for 1.55≤r≤10, while for r > 10, this condition may be satisfied. Further, we examine the validity of the energy constraints (15 ) graphically. From the plots of figure 18, it is seen that ρ + pr≥0 is not satisfied for all ζ values, while its counterpart in tangential pressure, i.e ρ + pt≥0, holds for all ζ. Collectively, one may conclude that the NEC is not valid in this case. Further, it is noticed that both inequalities, ρ − ∣pr∣≥0 and ρ − ∣pt∣≥0, are not valid for all − 5≤ζ≤10, however, ρ − ∣pt∣≥0 is true for some negative ζ values. Moreover, it is seen that ρ + pr − 2pt≥0, representing the inequality of the SEC, is valid for all − 5≤ζ≤10 values. Next, we examine the behavior of radial and tangential EoS parameters against r; the respective plots are provided in figure 19. It is seen that the EoS parameter for radial pressure exhibits positive behavior, the tangential EoS parameter shows negative behavior, while the anisotropic pressure ratio shows negative and linear conduct against r, as detailed in figure 20. In the right plot of same figure, it can be noticed that the anisotropy parameter exhibits negative behavior for either − 1 < ζ < 5 when r < 3 or − 1 < ζ < 0 when r > 3, hence leading to the attractive nature of gravity within the wormhole configuration.
Figure 17. Plot portrays the behavior of the shape model ( |
Figure 18. Graphs indicating the validity of energy constraints for the shape model ( |
Figure 19. Plots on left and right, respectively, refer to the behavior of EoS parameters for radial and tangential pressures. |
Figure 20. Graphs represent the dynamics of the pressure anisotropy ratio and anisotropy parameter Δ against r. |
For this wormhole shape function, the expression of the VIQ can be written as
$\begin{eqnarray*}VIQ=\frac{{\rm{e}}(-\frac{1}{{{\rm{e}}}^{2}}+\mathrm{ExpIntegral}E{\rm{i}}[-2]+\mathrm{ln}({r}_{0}))}{4(4\pi +\zeta )}-\frac{{\rm{e}}(-{{\rm{e}}}^{-\frac{2{r}_{1}}{{r}_{0}}}+\mathrm{ExpIntegral}E{\rm{i}}[-\frac{2{r}_{1}}{{r}_{0}}]+\mathrm{ln}({r}_{1})}{4(4\pi +\zeta )}.\end{eqnarray*}$
Also, three forces of the TOV equation can be computed as follows $\begin{eqnarray*}\begin{array}{rcl}{F}_{g} & = & \frac{{{\rm{e}}}^{1-\frac{2r}{{r}_{0}}}(2r+{r}_{0}+{{\rm{e}}}^{\frac{2r}{{r}_{0}}}{r}_{0})}{4{r}^{4}{r}_{0}(4\pi +\zeta )},\\ {F}_{h} & = & -\frac{{{\rm{e}}}^{-\frac{2r}{{r}_{0}}}\left(24{\rm{e}}\pi {r}_{0}(2r+3{r}_{0})+8{{\rm{e}}}^{\frac{2r}{{r}_{0}}}r{{r}_{0}}^{2}\zeta +{\rm{e}}(20{r}^{2}+42r{r}_{0}+33{{r}_{0}}^{2})\zeta +3{{\rm{e}}}^{1+\frac{2r}{{r}_{0}}}{{r}_{0}}^{2}(24\pi +11\zeta )\right)}{48{r}^{4}{{r}_{0}}^{2}(2\pi +\zeta )(4\pi +\zeta )},\\ {F}_{a} & = & \frac{{\rm{e}}{r}_{0}+2r{r}_{0}+{{\rm{e}}}^{1-\frac{2r}{{r}_{0}}}(2r+{r}_{0})}{2{r}^{4}{r}_{0}(4\pi +\zeta )}.\end{array}\end{eqnarray*}$
Moreover, the complexity factor is given by $\begin{eqnarray*}\begin{array}{rcl}{Y}_{TF} & = & -\frac{{\rm{e}}{r}_{0}+2r{r}_{0}+{{\rm{e}}}^{1-\frac{2r}{{r}_{0}}}(2r+{r}_{0})}{4{r}^{3}{r}_{0}(4\pi +\zeta )}-\frac{1}{2{r}^{3}}\\ & & \times \,\left[\frac{-8r{r}_{0}\zeta -2{{\rm{e}}}^{1-\frac{2r}{{r}_{0}}}(24\pi r+36\pi {r}_{0}+7r\zeta +11{r}_{0}\zeta )+3{\rm{e}}{r}_{0}\zeta \mathrm{ExpIntegral}E{\rm{i}}[-\frac{2r}{{r}_{0}}]+3{\rm{e}}{r}_{0}\zeta \mathrm{ln}(r)}{48{r}_{0}(2\pi +\zeta )(4\pi +\zeta )}\right.\\ & & -\,\left.\frac{-8{r}_{0}{r}_{2}\zeta -2{{\rm{e}}}^{1-\frac{2{r}_{2}}{{r}_{0}}}(36\pi {r}_{0}+24\pi {r}_{2}+11{r}_{0}\zeta +7{r}_{2}\zeta )+3{\rm{e}}{r}_{0}\zeta \mathrm{ExpIntegral}E{\rm{i}}[-\frac{2{r}_{2}}{{r}_{0}}]+3{\rm{e}}{r}_{0}\zeta \mathrm{ln}({r}_{2})}{48{r}_{0}(2\pi +\zeta )(4\pi +\zeta )}\right].\end{array}\end{eqnarray*}$
The active mass function can be computed as follows $\begin{eqnarray*}{{\mathbb{M}}}_{A}=-\frac{-\frac{{\rm{e}}{r}_{0}\zeta -8r{r}_{0}\zeta +{{\rm{e}}}^{1-(2r)/{r}_{0}}(96\pi r+26r\zeta +{r}_{0}\zeta )}{2{r}^{2}}-\frac{2{\rm{e}}(48\pi +13\zeta )\mathrm{ExpIntegral}E{\rm{i}}[-(2r)/{r}_{0}]}{{r}_{0}}}{48{r}_{0}(2\pi +\zeta )(4\pi +\zeta )}{| }_{{r}_{2}}^{R}.\end{eqnarray*}$
The graphical behavior of these quantities is provided in the plots of figures 21 and 22. It can be easily checked that the VIQ takes negative values and is in accordance with the limit r2 → r0 = 1.55 ⇒ VIQ → 0. The right plot of figure 21 witnesses the stability of the wormhole geometry as all forces almost balance each other's impact and vanish as r → 10. The left plot of figure 22 indicates that the exoticity factor takes negative for − 2≤ζ≤10 and − 10≤ζ≤ − 7; it hence can be concluded that exotic material is present in the neighborhood of the wormhole's throat, but that far from it, normal matter is present. The complexity factor also exhibits negative behavior, as shown in its right graph, and agrees with limit YTF → 0 as r → ∞. The mass ${{\mathbb{M}}}_{A}$ takes both positive and negative values for different choices of ζ, as shown in the left plot of figure 23, while its right plot witnesses the unstable behavior of wormhole configuration through the adiabatic index Γ, except for a few ζ choices where ${\rm{\Gamma }}\gt \frac{4}{3}$ can be attained.
Figure 21. Plots show the dynamics of the VIQ and stability of wormhole geometry through the TOV equation. |
Figure 22. Graphs provide the graphical illustration of exoticity and complexity factors against r. |
Figure 23. Plots refer to the dynamics of mass ${{\mathbb{M}}}_{A}$ and the adiabatic index versus r. |
4. Final remarks
In the literature, the construction of feasible wormhole geometry has been carried out in different gravitational frameworks and numerous captivating outcomes have been achieved so far. In this respect, the use of Casimir density has caught the special attention of researchers recently as it can provide a possibility to get exotic material artificially (in labs) and hence fulfills one of the crucial requirements for wormhole construction. This study will be a new and promising addition in this respect, where we have proposed some interesting wormhole models by taking the Yukawa-corrected Casimir density profile into account. To achieve this target, we have considered a well-motivated gravitational framework involving the interaction of curvature and matter, i.e. the ${\mathbb{F}}(R,{ \mathcal T })$ theory with ${L}_{m}={\mathbb{P}}$. For simplicity purposes, we have assumed a linear form of the ${\mathbb{F}}(R,{ \mathcal T })$ function, which is defined as ${\mathbb{F}}(R,{ \mathcal T })=R+2\zeta { \mathcal T }$, and anisotropic fluid as a source of ordinary matter. The work has been completed in three segments. In the first part, we have solved field equations using Yukawa-corrected Casimir density for shape function and explored its different features through graphs of some important quantities. In the second part, we have utilized the conformal symmetries in field equations and solved for the shape model by using Yukawa-corrected Casimir density. In the last case, we have considered a newly-proposed wormhole shape function in the ${\mathbb{F}}(R,{ \mathcal T })$ framework and solved the respective field equations for state variables, and also explored the significance of this shape model through graphs of the same quantities.
In the graphical analysis, we have explored the impact of the coupling parameter ζ on the resulting wormhole geometry. After a careful analysis of shape function S(r) and energy constraints in the first two cases, it is observed that the behavior of the obtained models is quite similar when variations of other parameters η, γ and σ are taken into account, while keeping ζ as a fixed quantity. Thus, only the ζ parameter has a strong impact on the solutions as for − 1 < ζ, the shape function exhibits decreasing behavior (with negative values of $S^{\prime} (r)$), while for ζ < − 1, the shape function shows increasing behavior with $0\lt S^{\prime} (r)\lt 1$. Below is a brief summary of our achieved outcomes:
The obtained results have also been summarized in the form of table 1. In the literature, numerous studies have been carried out by involving the Casimir effect and other matter sources in wormhole construction. Study of wormhole structures has been done in three ways: either by picking some specific choice of shape function [77-79] and exploring other parameters; by involving some EoS parameter [43, 82], or by utilizing some energy density profile [80, 100]. Since the last two techniques provide the exact form of wormhole shape function computed through field equations of MGFs, these techniques are therefore more promising.
Table 1. Summary of obtained results for wormhole solutions in ${\mathbb{F}}(R,{ \mathcal T })$ gravity. |
| Measure | Yukawa-Corrected Casimir Wormhole | Yukawa-Corrected Casimir Wormhole | New Wormhole Shape Model |
|---|---|---|---|
| without CKVs | with CKVs | with CKVs | |
| S(r) | Positive | Positive | Positive |
| decreasing for − 1 < ζ | increasing for ζ≤ − 1 | decreasing for all ζ | |
| $S^{\prime} ({r}_{1})\lt 1$ | Valid | Valid | Valid |
| $1-\frac{S}{r}\to 1$ | invalid as $1-\frac{S}{r}\approx 0.8$ | invalid as $1-\frac{S}{r}\approx 0.9$ | invalid as $1-\frac{S}{r}\approx 0.9$ |
| S(r) − r < 0 | Valid | Valid | Valid |
| ρ + pt | ρ + pt > 0 ∀ ζ | ρ + pt > 0 ∀ ζ | ρ + pt > 0 ∀ ζ |
| ρ + pr | ρ + pr < 0 ∀ ζ | ρ + pr < 0 ∀ ζ | ρ + pr < 0 ∀ ζ |
| VIQ | Negative, VIQ → 0 as r2 → r1 | Negative, VIQ → 0 as r2 → r1 | Negative, VIQ → 0 as r2 → r1 |
| ∀ ζ > − 1 | ∀ ζ > − 1 | ∀ ζ > − 1 | |
| Stability | Stable for all r | Stable for large r | Stable for large r |
| Complexity factor | Negative, YTF → 0 as r → 0 | Negative, YTF → 0 as r → 0 | Negative, YTF → 0 as r → 0 |
| (YTF) | ∀ ζ > − 1 | ∀ ζ > − 1 | ∀ ζ > − 1 |
| Adiabatic Index | Stable as Γ > 1.33 | Stable as Γ > 1.33 | Stable as Γ > 1.33 |
| − 2≤ζ≤10 | − 2≤ζ≤10 | − 2≤ζ≤10 | |
| Anisotropy Ratio | Δ < 0 for − 2≤ζ≤10 | Δ < 0 for − 2≤ζ≤10 | Δ < 0 for − 1≤ζ≤5, r < 3 |
| attractive nature of gravity | attractive nature of gravity | attractive nature of gravity | |
| Exoticity Factor ϒ | ϒ < 0 | ϒ < 0 | ϒ < 0 for − 10≤ζ≤ − 7 |
| for − 10≤ζ≤10 | for − 8≤ζ≤10 | and − 4≤ζ≤10 |
In the present work, our main goal is to assume matter density and explore the corresponding analytical form of shape function and other parameters. It is argued that the involvement of Casimir density in wormhole construction is of great interest as it can provide the only artificial source of exotic material needed to keep the wormhole open and stable. In this respect, Tripathy [58] discussed wormhole solutions by taking Casimir energy density in ${\mathbb{F}}(R,{ \mathcal T })$ theory (zero tidal force condition) and exploring the physical validity of solutions. In the present work, we have involved the Yukawa-correction term in Casimir density. An interesting feature of our work is the use of CKVs which leads to non-zero red-shift function. Thus, our work is more comprehensive in comparison to this work. Jawad et al [73] explored the wormhole solutions by involving a Yukawa-corrected form of Casimir density in f(T) gravity and found that their obtained solutions are stable but do not meet the asymptotically flat property. In another recent work [100], Casimir density has been applied to ${\mathbb{F}}(R,{ \mathcal T })$ theory by taking CKVs into account. In the majority of these works, only a limited number of parameters are explored to check the validity of solutions. Our work is quite general, in the sense that we have performed a very detailed graphical analysis of the presented models. For example, we have explored the stability of solutions through the TOV equation as well as the adiabatic index. Also, different measures—namely the exoticity factor, the anisotropy ratio, active gravitational mass, the complexity factor and the VIQ—are discussed in detail for the developed wormhole models. It is seen that all these measures show physically valid behavior, as required.
In the future, we would like to extend this work by considering other modified theories like the Gauss-Bonnet theory or Rastall theory and see the impact of the model parameters on the viability of wormhole geometry.